Polynomial and Radical Equations and Inequalities. Inequalities. Notes

Transcription

1 Notes Introduction In this unit, students etend their knowledge of first-degree equations and their graphs to radical equations and inequalities. Then they graph quadratic functions and solve quadratic equations and inequalities by various methods, including completing the square and using the Quadratic Formula. The unit concludes with methods for evaluating polynomial functions, including the Remainder and Factor Theorems. Students graph polynomial functions and investigate their roots and zeros. Finally, they study the composition of two functions, and then find the inverse of a function. Assessment Options Unit Test Pages 9 0 of the Chapter 7 Resource Masters may be used as a test or review for Unit. This assessment contains both multiple-choice and short answer items. Equations that model real-world data allow you to make predictions about the future. In this unit, you will learn about nonlinear equations, including polynomial and radical equations, and inequalities. Polynomial and Radical Equations and Inequalities Chapter Polynomials Chapter 6 Quadratic Functions and Inequalities Chapter 7 Polynomial Functions TestCheck and Worksheet Builder This CD-ROM can be used to create additional unit tests and review worksheets. 8 Unit Polynomial and Radical Equations and Inequalities 8 Unit Polynomial and Radical Equations and Inequalities

2 Teaching Suggestions Population Eplosion The United Nations estimated that the world s population reached 6 billion in 999. The population had doubled in about 0 years and gained billion people in just years. Assuming middle-range birth and death trends, world population is epected to eceed 9 billion by 00, with most of the increase in countries that are less economically developed. In this project, you will use quadratic and polynomial mathematical models that will help you to project future populations. Log on to Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit. Lesson Page USA TODAY Snapshots Tokyo leads population giants The 0 most populous urban areas in the world: Tokyo Meico City Bombay, India Sao Paulo, Brazil New York Lagos, Nigeria Los Angeles Shanghai, China Calcutta, India Buenos Aires, Argentina Source: United Nations (Millions) By Bob Laird, USA TODAY Have students study the USA TODAY Snapshot. Have students make conjectures about why a quadratic or polynomial model may be better than a linear one for modeling population data. According to the given data, what urban area has a population nearly as great as that of New York and Los Angeles combined? Tokyo Additional USA TODAY Snapshots appearing in Unit : Chapter Hanging on to the old buggy (p. 8) Chapter 6 More Americans study abroad (p. 8) Chapter 7 Digital book sales epected to grow (p. 68) Unit Polynomial and Radical Equations and Inequalities 9 Internet Project A WebQuest is an online project in which students do research on the Internet, gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance to the net step in their WebQuest. At the end of Chapter 7, the project culminates with a presentation of their findings. Teaching suggestions and sample answers are available in the WebQuest and Project Resources. Unit Polynomial and Radical Equations and Inequalities 9

3 Polynomials Chapter Overview and Pacing PACING (days) Regular Block Basic/ Basic/ Average Advanced Average Advanced Monomials (pp. 8) Multiply and divide monomials. Use epressions written in scientific notation. LESSON OBJECTIVES Polynomials (pp. 9 ) Add and subtract polynomials. Multiply polynomials. Dividing Polynomials (pp. 8) Divide polynomials using long division. Divide polynomials using synthetic division. Factoring Polynomials (pp. 9 ) Factor polynomials. Simplify polynomial quotients by factoring. Roots of Real Numbers (pp. 9) Simplify radicals. Use a calculator to approimate radicals. Radical Epressions (pp. 0 6) Simplify radical epressions. Add, subtract, multiply, and divide radical epressions. Rational Eponents (pp. 7 6) Write epressions with rational eponents in radical form, and vice versa. Simplify epressions in eponential or radical form. Radical Equations and Inequalities (pp. 6 69) Solve equations containing radicals. (with -8 Solve inequalities containing radicals. Follow-Up) Follow-Up: Solving Radical Equations and Inequalities by Graphing Comple Numbers (pp. 70 7) Add and subtract comple numbers. Multiply and divide comple numbers. Study Guide and Practice Test (pp. 76 8) Standardized Test Practice (pp. 8 8) Chapter Assessment TOTAL Pacing suggestions for the entire year can be found on pages T0 T. 0A Chapter Polynomials

4 Timesaving Tools Chapter Resource Manager All-In-One Planner and Resource Center See pages T T. CHAPTER RESOURCE MASTERS Study Guide and Intervention Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Applications* -Minute Check Transparencies 9 0 SM Interactive Chalkboard AlgePASS: Tutorial Plus (lessons) Materials SC algebra tiles GCS algebra tiles, graphing calculator , inde cards, string, small weights GCS (Follow-Up: graphing calculator) SC grid paper 9 06, 0 *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual Chapter Polynomials 0B

5 Mathematical Connections and Background Continuity of Instruction Prior Knowledge Students are familiar with the coefficients, variables, and positive integer eponents that make up monomials, and they have worked with radicals as square roots. They are familiar with the basic arithmetic operations (addition, subtraction, multiplication, division) that will be applied in new situations in this chapter. This Chapter Students learn how to apply the basic arithmetic operations to polynomials, radical epressions, and comple numbers. They eplore factoring polynomials and solving radical equations and inequalities. They learn how to write equivalent statements by using the Distributive Property, properties of eponents, or properties of radicals. They solve equations and inequalities involving polynomials, radicals, or comple numbers. Future Connections As early as the net two chapters, students will factor polynomials to solve quadratic equations, use comple numbers to epress the solution to equations or inequalities, and relate comple numbers and roots of polynomials. In their eploration of comple numbers, they will greatly epand on this chapter s brief introduction to the comple coordinate plane. Monomials Throughout this chapter students are introduced to new symbols and new notation. Each time, the new symbols are related to each other and to familiar ideas. In this lesson the new symbol is a negative eponent. The familiar idea is simplifying an epression, and to simplify a monomial means to write an equivalent epression without negative eponents and without parentheses. The rules for writing equivalent epressions include the definition of negative eponents and properties of eponents. The lesson introduces the terms coefficient and degree when discussing monomials. It also eplores writing and operating on numbers written in scientific notation, and using dimensional analysis to compute with units of measure. Polynomials In this lesson students find the sum or difference of several monomials, which is called a polynomial. The familiar ideas in this lesson are the operations of addition, subtraction, and multiplication as applied to polynomials. To add polynomials means to rewrite an indicated sum of polynomials as a sum of terms and then, by combining like terms, to rewrite that sum as a single polynomial. To subtract polynomials means to use the Distributive Property to rewrite the subtraction as a sum of terms, and then to combine like terms. To multiply a polynomial by a monomial means to use the Distributive Property to rewrite the product as a single polynomial. To multiply two binomials, the lesson shows how two applications of the Distributive Property result in finding the products of the First, Outer, Inner, and Last pairs of terms of the binomials. The two-time application of the Distributive Property is called the FOIL method. Dividing Polynomials This lesson eplores polynomial division; with the previous lesson, the four basic arithmetic operations are interpreted for polynomials. Dividing by a monomial uses the Distributive Property. Dividing a polynomial by a binomial (or by any polynomial) uses a process and format analogous to the long division algorithm for whole numbers. The student follows the four steps of divide, multiply, subtract, bring down ; the steps are repeated until there are no more terms in the dividend. The lesson also introduces an abbreviated form, called synthetic division, that records and manipulates just the coefficients of the polynomial terms. The divisor must be a binomial of degree, the terms of the dividend must be in descending order, using zeros to represent any missing terms, and the polynomial quotient must be written so that the leading coefficient of the divisor is. 0C Chapter Polynomials

6 Factoring Polynomials In factoring, a polynomial of degree two or more is rewritten as a product of polynomials each having a lesser degree. Taking out a common factor uses the Distributive Property; factoring by grouping uses two applications of the Distributive Property. Binomials written in the form of the difference of two squares or as the sum or difference of two cubes can be rewritten as a product of two factors, and a perfect square trinomial can be rewritten as the square of a binomial. Some other trinomials can be factored as the product of two binomials. Students reduce quotients of polynomials by removing common factors in the numerator and denominator. A record is kept of values of variables that would imply division by zero. Roots of Real Numbers This lesson begins with the familiar idea of a square root: a is a square root of b if a b. Then two new ideas are introduced. One idea is to use the same kind of definition to introduce the nth root of a number: a is an nth root of b if a n b. The second new idea is to introduce the symbols for principal roots. The principal square root is always a nonnegative number. The value of a principal nth root depends on the sign of the radicand and whether its inde is even or odd. The lesson eplores how to use absolute value symbols to simplify nth roots. Radical Epressions This lesson eplains that simplifying, as it pertains to radical epressions, takes into account the inde of the radical and the form of the radicand. The lesson also presents some rules for writing equivalent radical epressions, and students apply the rules as they add, subtract, multiply, and divide radicals. Rational Eponents The new symbol introduced in this lesson is a fraction used as an eponent. The rules for writing equivalent epressions for rational eponents include properties that describe how to translate between radical form and eponential form. Those rules include dealing with rational eponents that are unit fractions, either positive or negative. The rules for dealing with a base raised to a fractional eponent require that the denominator of the fraction is a positive integer and take into account the sign of the radicand and whether its inde is even or odd. Radical Equations and Inequalities This lesson deals with the familiar skills of solving equations and inequalities; the new concept is that the equations contain a variable inside a radical. No new properties are needed to solve these equations or inequalities; equivalent equations or inequalities are written until the variable is isolated on one side. At least once in the solution, both sides of the equation or inequality are raised to a power in order to remove a radical symbol. A most-important idea in the lesson is that sometimes this process of raising both sides to a power does not produce an equivalent statement. For eample, it is clear that has no real number solution. Squaring both sides results in ; the two statements are not equivalent and is not a solution to the original equation. Raising both sides to a power can introduce an etraneous solution, which is an apparent solution that will not satisfy the original equation or inequality. Comple Numbers This lesson introduces not simply a new symbol but a new set of numbers that are not part of the real number system. The new symbol is i, and a new rule is that an epression such as can be rewritten as the equivalent epression i. The comple number a bi can be treated as if it is a binomial, and operations on comple numbers follow the properties for adding, subtracting, multiplying, and dividing binomials, with one eception. That eception is to replace i with whenever i appears in an epression. Additional mathematical information and teaching notes are available in Glencoe s Algebra Key Concepts: Mathematical Background and Teaching Notes, which is available at The lessons appropriate for this chapter are as follows. Multiplying Monomials (Lesson ) Dividing Monomials (Lesson ) Adding and Subtracting Polynomials (Lesson ) Multiplying a Polynomial by a Monomial (Lesson ) Multiplying Polynomials (Lesson 6) Chapter Polynomials 0D

7 and Assessment Type Student Edition Teacher Resources Technology/Internet ASSESSMENT INTERVENTION Ongoing Prerequisite Skills, pp., 8,, 8,, 9, 6, 6, 67 Practice Quiz, p. 8 Practice Quiz, p. 6 Mied Review Error Analysis Standardized Test Practice Open-Ended Assessment Chapter Assessment pp. 8,, 8,, 9, 6, 6, 67, 7 Find the Error, pp. 6, 6 pp. 8,,, 6, 8,, 9,, 6, 67, 7, 8, 8 8 Writing in Math, pp. 7,, 8,, 9,, 6, 67, 7 Open Ended, pp. 6,, 6,, 7,, 60, 6, 7 Study Guide, pp Practice Test, p. 8 -Minute Check Transparencies Quizzes, CRM pp Mid-Chapter Test, CRM p. 09 Study Guide and Intervention, CRM pp. 9 0, 6,, 7 8, 6 6, 69 70, 7 76, 8 8, Cumulative Review, CRM p. 0 Find the Error, TWE pp. 6, 6 Unlocking Misconceptions, TWE pp.,,, 6,, 8 Tips for New Teachers, TWE pp. 8, 8,, 6, 6, 6, 67, 7 TWE p. Standardized Test Practice, CRM pp. Modeling: TWE pp., 9 Speaking: TWE pp. 8, 6, 6, 7 Writing: TWE pp., 8, 67 Open-Ended Assessment, CRM p. 0 Multiple-Choice Tests (Forms, A, B), CRM pp Free-Response Tests (Forms C, D, ), CRM pp Vocabulary Test/Review, CRM p. 06 AlgePASS: Tutorial Plus Standardized Test Practice CD-ROM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabulary_review Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review s Cracking the SAT & PSAT The Princeton Review s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment fleibility: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. 0E Chapter Polynomials

8 Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra Lesson AlgePASS Lesson - 8 Factoring Epressions II -8 9 Solving Radical Equations ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Log on for student study help. For each lesson in the Student Edition, there are Etra Eamples and Self-Check Quizzes. For chapter review, there is vocabulary review, test practice, and standardized test practice For more information on Intervention and Assessment, see pp. T8 T. Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Study Organizer, p. Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 6,, 6,, 7,, 60, 6, 7, 80) Writing in Math questions in every lesson, pp. 7,, 8,, 9,, 6, 67, 7 Reading Study Tip, pp. 9, 6,, 70, 7, 7 WebQuest, p. 7 Teacher Wraparound Edition Foldables Study Organizer, pp., 76 Study Notebook suggestions, pp. 6, 0, 6,, 7,, 60, 6, 7 Modeling activities, pp., 9 Speaking activities, pp. 8, 6, 6, 7 Writing activities, pp., 8, 67 Differentiated Instruction, (Verbal/Linguistic), p. 7 ELL Resources, pp. 0, 7,, 7,, 8,, 6, 66, 7, 7, 76 Additional Resources Vocabulary Builder worksheets require students to define and give eamples for key vocabulary terms as they progress through the chapter. (Chapter Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter Resource Masters, pp., 9,, 6, 67, 7, 79, 8, 9) Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Chapter Polynomials 0F

9 Notes Polynomials Have students read over the list of objectives and make a list of any words with which they are not familiar. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson. Lessons - through - Add, subtract, multiply, divide, and factor polynomials. Lessons - through -8 Simplify and solve equations involving roots, radicals, and rational eponents. Lesson -9 Perform operations with comple numbers. Many formulas involve polynomials and/or square roots. For eample, equations involving speeds or velocities of objects are often written with square roots. You can use such an equation to find the velocity of a roller coaster. You will use an equation relating the velocity of a roller coaster and the height of a hill in Lesson -6. Key Vocabulary scientific notation (p. ) polynomial (p. 9) FOIL method (p. 0) synthetic division (p. ) comple number (p. 7) NCTM Local Lesson Standards Objectives -,, 6, 7, 8, 9, 0 -,, 6, 8, 9, 0 -,, 6, 7, 8, 9 -,,, 6, 8, 9, 0 -,, 6, 7, 8, 9-6,, 6, 7, 8, 9, 0-7,, 6, 8, 9-8,, 6, 8, 9, 0-8,, 0 Follow-Up -9,,, 6, 7, 8, 9, 0 Key to NCTM Standards: =Number & Operations, =Algebra, =Geometry, =Measurement, =Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 0=Representation 0 Chapter Polynomials Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter test. 0 Chapter Polynomials

10 Prerequisite Skills To be successful in this chapter, you ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter. For Lessons - and -9 Rewrite Differences as Sums Rewrite each difference as a sum.. 7 (7). 6 6 (). y (y). 8 8 (). y 6yz y (6yz) 6. 6a b b c 6a b (b c) For Lesson - Distributive Property Use the Distributive Property to rewrite each epression without parentheses. (For review, see Lesson -.) ( ) 8. ( ) 9. ( ) 0. ( ). (a ) a. ( 6z) z 6 6 For Lessons - and -9 Classify Numbers Find the value of each epression. Then name the sets of numbers to which each value belongs. (For review, see Lesson -.) 8. See margin () This section provides a review of the basic concepts needed before beginning Chapter. Page references are included for additional student help. Prerequisite Skills in the Getting Ready for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Prerequisite Lesson Skill - Distributive Property (p. 8) - Properties of Eponents (p. ) - Rational and Irrational Numbers (p. ) -6 Multiplying Binomials (p. 9) -8 Multiplying Radicals (p. 6) -9 Binomials (p. 67) Fold and Cut First Sheets Make this Foldable to record information about polynomials. Begin with four sheets of grid paper. Second Sheets Fold in half along the width. On the first two sheets, cut along the fold at the ends. On the second two sheets, cut in the center of the fold as shown. Fold and Label Insert first sheets through second sheets and align folds. Label pages with lesson numbers. Answers. 6.; reals, rationals. 6; reals, rationals, integers. 7; reals, rationals, integers, whole numbers, natural numbers 6. ; reals, irrationals 7. ; reals, rationals, integers, whole numbers, natural numbers 8. 6; reals, rationals, integers, whole numbers, natural numbers Reading and Writing As you read and study the chapter, fill the journal with notes, diagrams, and eamples for polynomials. Chapter Polynomials For more information about Foldables, see Teaching Mathematics with Foldables. TM Organization of Data and Journal Writing When labeling the pages for the lessons, combine Lessons - and - on the same page and Lessons -8 and -9 on the same page. Use etra pages for vocabulary lists and applications. Writer s journals can also be used to record the direction and progress of learning, to describe positive and negative eperiences during learning, to write about personal associations and eperiences while learning, and to list eamples of ways in which new knowledge has or will be used in their daily life. Chapter Polynomials

11 Lesson Notes Monomials Focus -Minute Check Transparency - Use as a quiz or a review of Chapter. Mathematical Background notes are available for this lesson on p. 0C. is scientific notation useful in economics? Ask students: What are the powers of ten?, 0, 0, 0, 0 0, 0, 0, 0, What are some other fields that use scientific notation for very large or very small numbers? astronomy, biology, computer science Teach MONOMIALS In-Class Eample Power Point Teaching Tip Help students think carefully about the meaning of eponents by asking them to read this epression aloud correctly. If students read as three, instead of correctly saying cubed or to the third (power), they are apt to confuse with. Simplify (a b)(ab ). 0a b Resource Manager Vocabulary monomial constant coefficient degree power simplify standard notation scientific notation dimensional analysis Chapter Polynomials Multiply and divide monomials. Use epressions written in scientific notation. MONOMIALS A monomial is an epression that is a number, a variable, or the product of a number and one or more variables. Monomials cannot contain variables in denominators, variables with eponents that are negative, or variables under radicals. Negative Eponents Words For any real number a 0 and any integer n, a n a n and a n an. Eamples and b 8 b8 To simplify an epression containing powers means to rewrite the epression without parentheses or negative eponents. Eample is scientific notation useful in economics? Economists often deal with very large numbers. For eample, the table shows the U.S. public debt for several years in the last century. Such numbers, written in standard notation, are difficult to work with because they contain so many digits. Scientific notation uses powers of ten to make very large or very small numbers more manageable. Monomials Not Monomials b, w,,, y n,, 8, a Constants are monomials that contain no variables, like or. The numerical factor of a monomial is the coefficient of the variable(s). For eample, the coefficient of m in 6m is 6. The degree of a monomial is the sum of the eponents of its variables. For eample, the degree of g 7 h is 7 or. The degree of a constant is 0. Apower is an epression of the form n. The word power is also used to refer to the eponent itself. Negative eponents are a way of epressing the multiplicative inverse of a number. For eample, can be written as. Note that an epression such as is not a monomial. Why? Simplify Epressions with Multiplication Simplify ( y )( y ). ( y )( y ) ( y y)( y y y y) Definition of eponents () y y y y y y y 6 U.S. Debt ($) $ $,00,000,000 $ $ $ $ 6,00,000,00 $ Public Debt $ $ $ 8,00,000,000 $ $ $ $ Source: U.S. Department of the Treasury $ $,,00,000,000 $ $ Commutative Property Definition of eponents,67,00,000,000 $ Year $ Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp. 9 0 Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. Enrichment, p. Science and Mathematics Lab Manual, pp. 09 Transparencies -Minute Check Transparency - Answer Key Transparencies Technology Interactive Chalkboard

12 Eample suggests the following property of eponents. Product of Powers Words For any real number a and integers m and n, a m a n a m n. Eamples 9 and b b b 8 To multiply powers of the same variable, add the eponents. Knowing this, it seems reasonable to epect that when dividing powers, you would subtract eponents. Consider 9. 9 Remember that 0. Simplify. Definition of eponents It appears that our conjecture is true. To divide powers of the same base, you subtract eponents. Simplify Epressions with Division Quotient of Powers Words For any real number a 0, and integers m and n, a m an a m n. Eamples or and 7 7 or Eample Simplify p p8. Assume that p 0. p p8 p 8 Subtract eponents. p or p Remember that a simplified epression cannot contain negative eponents. CHECK p p p p p8 p p p p p p p p Definition of eponents In-Class Eample s Power Point Simplify. Assume that s 0 s 0. s 8 Teaching Tip When discussing In-Class Eample, if any students get the incorrect answer, lead them to understand s that they divided the eponents instead of subtracting them as a method for dividing the two epressions. Interactive Chalkboard PowerPoint Presentations This CD-ROM is a customizable Microsoft PowerPoint presentation that includes: Step-by-step, dynamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The -Minute Check Transparencies Hot links to Glencoe Online Study Tools p Simplify. You can use the Quotient of Powers property and the definition of eponents to simplify y y, if y 0. Method Method y y y Quotient of Powers y y y y y y y y y y y 0 Subtract. Divide. Definition of eponents In order to make the results of these two methods consistent, we define y 0, where y 0. In other words, any nonzero number raised to the zero power is equal to. Notice that 0 0 is undefined. Lesson - Monomials Unlocking Misconceptions Correcting Errors Encourage students to analyze the error or errors they made when they get an incorrect answer. Stress that students should use errors as an opportunity to clarify their thinking. Using Definitions Suggest that students return to the basic definitions for eponents rather than just memorizing rules. For eample, they can derive the rule for multiplying quantities such as by rewriting the problem as. Lesson - Monomials

13 In-Class Eamples Simplify each epression. a. (b ) b 8 b. (c d ) 7c 6 d c. d. a b 8 a b 0 Power Point Teaching Tip Encourage students to begin each step of a simplification by naming the operation they are about to perform, such as finding the power of a power. a Simplify y. a 6y b a y b 0 Concept Check Monomials Have students write their own summary of the properties of eponents, such as to multiply epressions with eponents, you add the eponents; to divide, you subtract the eponents and so on. Study Tip Simplified Epressions A monomial epression is in simplified form when: there are no powers of powers, each base appears eactly once, all fractions are in simplest form, and there are no negative eponents. The properties we have presented can be used to verify the properties of powers that are listed below. Properties of Powers Words Suppose a and b are real numbers and m Eamples and n are integers. Then the following properties hold. Power of a Power: (a m ) n a mn (a ) a 6 Eample Power of a Product: (ab) m a m b m (y) y Power of a Quotient: a b n a n b, n b 0 and a b n b a n or b n, a 0, b 0 a n Simplify Epressions with Powers Simplify each epression. a. (a ) 6 b. (p s ) b a b a y y (a ) 6 a (6) Power of a power (p s ) () (p ) (s ) c. y a 8 p s 0 Power of a power d. a a a y ( y ) Power of a quotient Power of a quotient ( ) y Power of a product a Power of a quotient 8 y () 8 6 a 6 With complicated epressions, you often have a choice of which way to start simplifying. Eample Simplify Epressions Using Several Properties n. Simplify ny Method Method Raise the numerator and denominator Simplify the fraction before raising to the fourth power before simplifying. to the fourth power. n ny ( ( n ) n y) n ny ( ) ( n ) ( n) ( y) 6 n 8n y 6 n 8n y 6 n y n n y n y 6 n y Chapter Polynomials Chapter Polynomials

14 Study Tip Graphing Calculators To solve scientific notation problems on a graphing calculator, use the EE function. Enter as 6.8 nd [EE] 6. SCIENTIFIC NOTATION The form that you usually write numbers in is standard notation. A number is in scientific notation when it is in the form a 0 n, where a 0 and n is an integer. Scientific notation is used to epress very large or very small numbers. Eample Epress Numbers in Scientific Notation Epress each number in scientific notation. a. 6,80,000 6,80, ,000, Write,000,000 as a power of 0. b or 00, Use a negative eponent. You can use properties of powers to multiply and divide numbers in scientific notation. Eample 6 Multiply Numbers in Scientific Notation Evaluate. Epress the result in scientific notation. a. ( 0 )( 0 7 ) ( 0 )( 0 7 ) ( ) (0 0 7 ) Associative and Commutative Properties 8 0 8, or 0 b. (.7 0 )( 0 6 ) (.7 0 )( 0 6 ) (.7 ) (0 0 6 ) Associative and Commutative Properties , or 0 SCIENTIFIC NOTATION In-Class Eamples Epress each number in scientific notation. a.,60, b Power Point 6 Evaluate. Epress the result in scientific notation. a. ( 0 )(7 0 8 ). 0 b. (.8 0 )( 0 7 ) 7. 0 BIOLOGY There are about 0 6 red blood cells in one milliliter of blood. A certain blood sample contains red blood cells. About how many milliliters of blood are in the sample? about.66 ml Astronomy Light travels at a speed of about m/s. The distance that light travels in a year is called a light-year. Source: Real-world problems often involve units of measure. Performing operations with units is known as dimensional analysis. Eample 7 Divide Numbers in Scientific Notation ASTRONOMY After the Sun, the net-closest star to Earth is Alpha Centauri C, which is about 0 6 meters away. How long does it take light from Alpha Centauri C to reach Earth? Use the information at the left. Begin with the formula d rt, where d is distance, r is rate, and t is time. t d r Solve the formula for time. 0 6 m Distance from Alpha Centauri C to Earth m/s speed of light Estimate: The result should be slightly greater than 0 6 /s 08 or s. 00., or It takes about. 0 8 seconds or. years for light from Alpha Centauri C to reach Earth. Lesson - Monomials Differentiated Instruction Interpersonal Have students discuss with a partner or in a small group the methods for multiplying and dividing monomial epressions with eponents, and also numbers written in scientific notation. Ask them to work together to develop a list of common errors for such problems, and to suggest ways to correct and avoid these errors. Lesson - Monomials

15 Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. add the information about the meaning of dimensional analysis to their notebook. include any other item(s) that they find helpful in mastering the skills in this lesson. About the Eercises Organization by Objective Monomials: 8 Scientific Notation: 60 Odd/Even Assignments Eercises 8 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Eercise 9 involves research on the Internet or other reference materials. Assignment Guide Basic: 9 9 odd, 7 odd, 9 8 Average: 9 7 odd, 9 8 Advanced: 8 8 even, (optional: 79 8) FIND THE ERROR Suggest that students use two steps to simplify epressions such as by first rewriting with the () reciprocal and then squaring. Answers. Sample answer: ( ) 8 6 since ( ) ( ) ( ) ( ) 8 6 Concept Check. Sometimes; in general y z y z, so y z yz when y z yz, such as when y and z.. Alejandra; when Kyle used the Power of a Product property in his first step, he forgot to put an eponent of on a. Also, in his second step, () should be, not. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 9 0, 6 6, 7 7 Application indicates increased difficulty Practice and Apply Homework Help For See Eercises Eamples 8, , 9, 6, 7 0, 8, 9 6, 7 Etra Practice See page Chapter Polynomials. OPEN ENDED Write an eample that illustrates a property of powers. Then use multiplication or division to eplain why it is true. See margin.. Determine whether y z yz is sometimes, always, or never true. Eplain. a. FIND THE ERROR Alejandra and Kyle both simplified b. ( ab ) Alejandra a b ( ab) = (a b)( ab ) = (a b)( ) a (b ) = (a b) a b 6 = 8a b 7 Who is correct? Eplain your reasoning. Simplify. Assume that no variable equals (b) 6b 6. (n ) (n ) 0y 7. y 6y 8. a 8a b6 b a b 8p6q 9 9. ( p q) 9p q 0. w z w z 6. c d c 9 d Epress each number in scientific notation.., Evaluate. Epress the result in scientific notation.. (. 0 8 )(. 0 ) ASTRONOMY Refer to Eample 7 on page. The average distance from Earth to the Moon is about meters. How long would it take a radio signal traveling at the speed of light to cover that distance? about.8 s m Simplify. Assume that no variable equals a a 6 a 8 9. b b 7 b 0. (n ) n 6. (z ) z 0. () 6. (c) 8c. a n6 an an. y z7 y z y z 6. (7 y )(y ) 8 y 7. (b c)(7b c ) b c 8. (a b )(ab) ab 9. (r s) (rs ) r 7 s Kyle a b a ( ab ) = b ( ) a(b ) ab = ab 6 = a bb 6 a = a b 7 0. (6y )( y) y. a(a b)(6ab ) 90a b. yz 0 y7z y m factors m factors m factors 6. (ab) m ab ab ab a a a b b b a m b m 6. Economics often involves large amounts of money. Answers should include the following. The national debt in 000 was five trillion, si hundred seventy-four billion, two hundred million or.67 0 dollars. The population was two hundred eighty-one million or Divide the national debt by the population: $.09 0 or about $0,9 per person. 6 Chapter Polynomials

16 A scatter plot of populations will help you make a model for the data. Visit webquest to continue work on your WebQuest project.. If r r, what is the value of r? 6. What value of r makes y 8 y r y 7 true? 7 ab 9a c b7c a c b. c d( 0c cd ) d cd. m n 8 6m ( m n ) n m n ab 6a a b 6b 7. 6 y y 8 y 6 8. y y 9. v w v w a b6 60a 6 b 8 a b. yz8 0 6 y z y z7 Epress each number in scientific notation , ,00, ,00, Evaluate. Epress the result in scientific notation. 0. (. 0 )( ) (.0 0 )( 0 ) (6. 0 ) (. 0 ) POPULATION The population of Earth is about 6,080,000,000. Write this number in scientific notation BIOLOGY Use the diagram at the right to write the diameter of a typical flu virus in scientific notation. 0 7 m 8. CHEMISTRY One gram of water contains about. 0 molecules. About how many molecules are in 00 grams of water? m RESEARCH Use the Internet or other source to find the masses of Earth and the Sun. About how many times as large as Earth is the Sun? about 0,000 times 60. CRITICAL THINKING Determine which is greater, 00 0 or Eplain (0 ) 0 or 0 0, and , so CRITICAL THINKING For Eercises 6 and 6, use the following proof of the Power of a Power Property. 6. Definition of an eponent m factors n factors a m a n a a a a a a m n factors a a a a m n 6. What definition or property allows you to make each step of the proof? 6. Prove the Power of a Product Property, (ab) m a m b m. See margin. 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. Why is scientific notation useful in economics? Include the following in your answer: the 000 national debt of $,67,00,000,000 and the U.S. population of 8,000,000, both written in words and in scientific notation, and an eplanation of how to find the amount of debt per person, with the result written in scientific notation and in standard notation. Lesson - Monomials 7 Study Study Guide and and Intervention Intervention, p. 9 (shown) and p. 0 - NAME DATE PERIOD Monomials Monomials A monomial is a number, a variable, or the product of a number and one or more variables. Constants are monomials that contain no variables. Negative Eponent a n and a n for any real number a 0 and any integer n. a n a n When you simplify an epression, you rewrite it without parentheses or negative eponents. The following properties are useful when simplifying epressions. Product of Powers a m a n a m n for any real number a and integers m and n. a Quotient of Powers m a m n for any real number a 0 and integers m and n. a n For a, b real numbers and m, n integers: (a m ) n a mn (ab) m a m b m Properties of Powers a n a n, b 0 b b n a n n b b or n, a 0, b 0 b a a n Eample Simplify. Assume that no variable equals 0. a. (m n )(mn) (m b. ) (m (m n )(mn) m n m n ) 7m m n n (m ) m 7m n (m ) m 7m 6 m m m 6 Eercises Simplify. Assume that no variable equals 0.. c c c 6 c b. 8 b 6. (a ) a 0 b y 6. y. a b y a b 6. y 8m n mn 7. (a b ) (abc) a 6 b 8 c 8. m 7 m 8 m 9. m n 0. c t. j(j k )(j k 7 j mn ). (m n) m c t k m n - b a NAME 9 DATE PERIOD Practice (Average) Skills Practice, p. and Practice, p. (shown) Simplify. Assume that no variable equals 0.. n n n 7. y 7 y y y. t 9 t 8 t.. (f ) 6 6f 6. (b c ) 8c 9 b 6 7. (d t v )(dt v 0d ) t 8. 8u(z) 6uz v m 9. 8 y m s 7 y 6s my 8s 7 7 ( 7 ) 6... (w z )(8w) 6. (m n 6 ) (m n p ) 6 m n 6 p 0 wz Monomials r s z 6 y y. d f d f d f r s 6 z ( 7. y )(y 8 ) 0(m v 8. v)(v) ( ) y y (v) (m ) m Epress each number in scientific notation , Evaluate. Epress the result in scientific notation.. (.8 0 )(6.9 0 ). ( )( ) COMPUTING The term bit, short for binary digit, was first used in 96 by John Tukey. A single bit holds a zero or a one. Some computers use -bit numbers, or strings of consecutive bits, to identify each address in their memories. Each -bit number corresponds to a number in our base-ten system. The largest -bit number is nearly,9,000,000. Write this number in scientific notation LIGHT When light passes through water, its velocity is reduced by %. If the speed of light in a vacuum is.86 0 miles per second, at what velocity does it travel through water? Write your answer in scientific notation..9 0 mi/s 7. TREES Deciduous and coniferous trees are hard to distinguish in a black-and-white photo. But because deciduous trees reflect infrared energy better than coniferous trees, the two types of trees are more distinguishable in an infrared photo. If an infrared wavelength measures about meters and a blue wavelength measures about. 0 7 meters, about how many times longer is the infrared wavelength than the blue wavelength? about.8 times Reading to Learn Mathematics, p. - NAME DATE PERIOD y 6 Reading to Learn Mathematics Monomials Pre-Activity Why is scientific notation useful in economics? Read the introduction to Lesson - at the top of page in your tetbook. Your tetbook gives the U.S. public debt as an eample from economics that involves large numbers that are difficult to work with when written in standard notation. Give an eample from science that involves very large numbers and one that involves very small numbers. Sample answer: distances between Earth and the stars, sizes of molecules and atoms Reading the Lesson. Tell whether each epression is a monomial or not a monomial. If it is a monomial, tell whether it is a constant or not a constant. a. monomial; not a constant b. y y 6 not a monomial c. 7 monomial; constant d. z not a monomial y y ELL Lesson - Lesson - NAME DATE PERIOD - Enrichment, p. Properties of Eponents The rules about powers and eponents are usually given with letters such as m, n, and k to represent eponents. For eample, one rule states that a m a n a m n. In practice, such eponents are handled as algebraic epressions and the rules of algebra apply. Eample Simplify a (a n a n ). a (a n a n ) a a n a a n Use the Distributive Law. a n a n Recall a m a n a m n. a n a n Simplify the eponent n as n. It is important always to collect like terms only. Eample Simplify (a n b m ). (a n b m ) (a n b m )(a n b m ) F O I L b a n a n m a n m b m e like ter. Complete the following definitions of a negative eponent and a zero eponent. For any real number a 0 and any integer n, a n a n. For any real number a 0, a 0.. Name the property or properties of eponents that you would use to simplify each epression. (Do not actually simplify.) m a. 8 quotient of powers m b. y 6 y 9 product of powers c. (r s) power of a product and power of a power Helping You Remember. When writing a number in scientific notation, some students have trouble remembering when to use positive eponents and when to use negative ones. What is an easy way to remember this? Sample answer: Use a positive eponent if the number is 0 or greater. Use a negative number if the number is less than. Lesson - Monomials 7

17 Assess Open-Ended Assessment Speaking Have students eplain in informal language how to tell whether a number is written in scientific notation. Then have them eplain how to simplify monomial epressions involving negative eponents. Standardized Test Practice Maintain Your Skills Mied Review 6. Simplify ( ). D A B C D ? B A 7,000 B 70,000 C 7,00,000 D 7,000,000 Solve each system of equations by using inverse matrices. (Lesson -8) 66. y 8 (, ) 67. y 9 (, ) y y Intervention Simplifying epressions with eponents is a skill that is needed frequently in algebra. Take time to clear up any misconceptions at this point and to help students develop an understanding of the properties so that they remember the procedures correctly for later use. New Getting Ready for Lesson - PREREQUISITE SKILL Lesson - presents multiplying polynomials. This multiplication involves the use of the Distributive Property. Eercises 79 8 should be used to determine your students familiarity with the Distributive Property. 7. See margin. 7. Sample answer using (0,.9) and (8, 8.): y 0..9 Find the inverse of each matri, if it eists. (Lesson -7) Evaluate each determinant. (Lesson -) Solve each system of equations. (Lesson -) 7. y (,, ) 7. a b c 6 (, 0, ) y z a b c 6 y z a b c 6 TRANSPORTATION For Eercises 7 76, refer to the graph at the right. (Lesson -) 7. Make a scatter plot of the data, where the horizontal ais is the number of years since Write a prediction equation. 76. Predict the median age of vehicles on the road in 00. Sample answer: 9.7 yr USA TODAY Snapshots Hanging on to the old buggy The median age of automobiles and trucks on the road in the USA: Source: Transportation Department.9 years. years 6 years 6.9 years 6. years 7.7 years 8. years By Keith Simmons, USA TODAY Answer 7. Median Age (yr) Median Age of Vehicles y Years Since 970 Getting Ready for the Net Lesson 8 Chapter Polynomials Solve each equation. (Lesson -) Use the Distributive Property to find each product. (To review the Distributive Property, see Lesson -.) 79. ( y) y 80. ( z) z 8. ( ) 8 8. ( ) ( y) 0y 8. (y ) y Online Lesson Plans USA TODAY Education s Online site offers resources and interactive features connected to each day s newspaper. Eperience TODAY, USA TODAY s daily lesson plan, is available on the site and delivered daily to subscribers. This plan provides instruction for integrating USA TODAY graphics and key editorial features into your mathematics classroom. Log on to 8 Chapter Polynomials

18 Polynomials Lesson Notes Vocabulary polynomial terms like terms trinomial binomial FOIL method Study Tip Reading Math The prefi bi- means two, and the prefi tri- means three. Add and subtract polynomials. Multiply polynomials. Shenequa wants to attend Purdue University in Indiana, where the out-of-state tuition is $,87. Suppose the tuition increases at a rate of % per year. You can use polynomials to represent the increasing tuition costs. ADD AND SUBTRACT POLYNOMIALS If r represents the rate of increase of tuition, then the tuition for the second year will be,87( r). For the third year, it will be,87( r), or,87r 7,7r,87 in epanded form. The epression,87r 7,7r,87 is called a polynomial. A polynomial is a monomial or a sum of monomials. The monomials that make up a polynomial are called the terms of the polynomial. In a polynomial such as, the two monomials and can be combined because they are like terms. The result is. The polynomial is a trinomial because it has three unlike terms. A polynomial such as y z is a binomial because it has two unlike terms. The degree of a polynomial is the degree of the monomial with the greatest degree. For eample, the degree of is, and the degree of y z is. Eample Degree of a Polynomial Determine whether each epression is a polynomial. If it is a polynomial, state the degree of the polynomial. a. 6 y 9 This epression is a polynomial because each term is a monomial. The degree of the first term is or 8, and the degree of the second term is. The degree of the polynomial is 8. b. This epression is not a polynomial because is not a monomial. To simplify a polynomial means to perform the operations indicated and combine like terms. Eample can polynomials be applied to financial situations? Subtract and Simplify Simplify ( ) ( ). ( ) ( ) Distribute the. ( ) ( ) ( ) 6 Year Tuition $,87 $,7 $,00 $,60 Group like terms. Combine like terms. Lesson - Polynomials 9 Focus -Minute Check Transparency - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. 0C. can polynomials be applied to financial situations? Ask students: What is meant by tuition increases at a rate of % per year? Each year the tuition is % higher than it was the year before. Will the amount of the tuition increase be the same each year? no Teach ADD AND SUBTRACT POLYNOMIALS In-Class Eamples Power Point Determine whether each epression is a polynomial. If it is a polynomial, state the degree of the polynomial. a. c c 8 no b. 6p p q 7 yes, 9 Simplify (a a 7) (a a ). a 8a 9 Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp. 6 Skills Practice, p. 7 Practice, p. 8 Reading to Learn Mathematics, p. 9 Enrichment, p. 0 School-to-Career Masters, p. 9 Teaching Algebra With Manipulatives Masters, p. Transparencies -Minute Check Transparency - Answer Key Transparencies Technology Interactive Chalkboard Lesson - Lesson Title 9

19 MULTIPLY POLYNOMIALS In-Class Eamples Find y(y y ). y y y Find (p )(p ). 8p p Find (a a )(a ). a a a 8 Practice/Apply Study Notebook Power Point Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. include any other item(s) that they find helpful in mastering the skills in this lesson. About the Eercises Organization by Objective Add and Subtract Polynomials: 6 7 Multiply Polynomials: 8, 7 0 Odd/Even Assignments Eercises 6 and 7 0 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 7 odd,, 6, 7 odd,, 69 Average: 7 odd,, 6, 7 odd, 69 Advanced: 6 even,, 6, 8 even, 6 (optional: 66 69) Study Tip Vertical Method You may also want to use the vertical method to multiply polynomials. y () y y 8 y 0y y y 8 0 Chapter Polynomials Answers. 7. y y. r r 6. 0m m 6. y 6y MULTIPLY POLYNOMIALS polynomials. Eample Multiplying Binomials Use algebra tiles to find the product of and. Draw a 90 angle on your paper. Use an tile and a tile to mark off a length equal to along the top. Use the tiles to mark off a length equal to along the side. Draw lines to show the grid formed. Fill in the lines with the appropriate tiles to show the area product. The model shows the polynomial 7 0. The area of the rectangle is the product of its length and width. Substituting for the length, width, and area with the corresponding polynomials, we find that ( )( ) 7 0. FOIL Method for Multiplying Binomials The product of two binomials is the sum of the products of F the first terms, O the outer terms, I the inner terms, and L the last terms. Eample Multiply and Simplify Find (7 ). (7 ) (7 ) () () 6 0 Multiply Polynomials Find (n 6n )(n ). (n 6n )(n ) n (n ) 6n(n ) ()(n ) n n n 6n n 6n () n () n n 6n n n 8 n 0n n 8 You can use the Distributive Property to multiply Distributive Property Multiply the monomials. You can use algebra tiles to model the product of two binomials. In Eample, the FOIL method is used to multiply binomials. The FOIL method is an application of the Distributive Property that makes the multiplication easier. Eample Multiply Two Binomials Find (y )(y ). (y )(y ) y y y y First terms Outer terms Inner terms Last terms y y 8 Multiply monomials and add like terms. Algebra Activity Distributive Property Distributive Property Multiply monomials. Combine like terms. Materials: protractor, ruler/straightedge, algebra tiles Remind students that the length of an tile is not a multiple of the length of a side of a unit tile. Point out to students that the width of an tile is eactly one unit (the same as the length of a side of a unit tile). 0 Chapter Polynomials

20 Concept Check. Sample answer:. OPEN ENDED Write a polynomial of degree that has three terms.. Identify the degree of the polynomial 7.. Model ( ) using algebra tiles. See pp. 8A 8B. Study Study Guide and and Intervention Intervention, p. (shown) and p. 6 - NAME DATE PERIOD Polynomials Add and Subtract Polynomials Polynomial Like Terms a monomial or a sum of monomials terms that have the same variable(s) raised to the same power(s) To add or subtract polynomials, perform the indicated operations and combine like terms. Eample Simplify 6rs 8r s r 8rs 6s. 6rs 8r s r 8rs 6s (8r r ) (6rs 8rs) (s 6s ) Group like terms. r rs s Combine like terms. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7, 8 9, 0 Application Determine whether each epression is a polynomial. If it is a polynomial, state the degree of the polynomial.. a b yes,. 9y yes, 6. m w nz no Simplify p q 6p q 8p q 7. (a b) (8a b) 0a b 8. ( ) ( ) 9. (y 9) 6y 8 0. p q(pq p q pq ). (y 0)(y 7) y y 70. ( 6)( ) 9 8. (z )(z ) z. (m n) m mn 9n. GEOMETRY Find the area of the triangle. 7.. ft ft Eample Simplify y y 7 y (0y y 8 y). y y 7 y (0y y 8 y) y y 7 y 0y y 8 y Distribute the minus sign. (7 y 8 y ) (y y ) (y 0y) Group like terms. y y 8y Combine like terms. Eercises Simplify.. (6 ) ( ). (7y y ) (6y y ) y 8y 8. (m 6m) (6m m ). 7 y y 8m m 7y. (8p pq 6q ) (p pq q ) 6. 7j k j j k p pq 0q j k 7. (8m 7n ) (n m ) 8. bc 6b c 8b 8c 8bc 0m 8n b bc c 9. 6r s rs r s 7rs r s 9rs 0. 9y y (6y y ) r s rs y 0y. (y 8 y) ( 7y 8y). 0.8b.7b 7. (.9b.6b.) 7 y y 7.9b.b 0.. (bc 9b 6c ) (c b bc). y 6y y y 0 0b 8bc c y 7y Lesson y y y y 6. p p p p p 7p 7 y y 8 8 9p p p indicates increased difficulty Practice and Apply Homework Help For Eercises See Eamples 6 7,, 6, 8, 7, 8, 7 6,, 9, 0, Etra Practice See page Answers NAME DATE PERIOD - Enrichment, p. 0 Polynomials with Fractional Coefficients Polynomials may have fractional coefficients as long as there are no variables in the denominators. Computing with fractional coefficients is performed in the same way as computing with whole-number coefficients. Simpliply. Write all coefficients as fractions.. m 7 p n 7 p m n Lesson - Polynomials m 0 n p. y z y z y z 8. a ab b 6 a ab b a ab b ft Determine whether each epression is a polynomial. If it is a polynomial, state the degree of the polynomial. 6. z z yes, 7. 9 yes, 8. 6 y c no z d 9. m no 0. y yes, 6. y 6 y7 yes, 7 Simplify.. See margin.. ( ) ( 9). (y y ) (8y 6y ). (9r 6r 6) (8r 7r 0). (7m m 9) (m 6) 6. ( y y) (8y y ) 7. (0 y y ) ( y) 8. b(cb zd) 9. a(a b) 0. ab (a b 6a b a b ). y(y y y ). (8 y 6y ). a (a 6b 8ab ). PERSONAL FINANCE Toshiro wants to know how to invest the $80 he has saved. He can invest in a savings account that has an annual interest rate of.7%, and he can invest in a money market account that pays about.% per year. Write a polynomial to represent the amount of interest he will earn in year if he invests dollars in the savings account and the rest in the money market account E-SALES For Eercises and 6, use the following information. A small online retailer estimates that the cost, in dollars, associated with selling units of a particular product is given by the epression The revenue from selling units is given by Write a polynomial to represent the profit generated by the product. 6. Find the profit from sales of 80 units. $ y y 8. b c bdz 9. a ab 0. a b 0a b a b 6. 6 y 8 y y. 6 9 y y. a a b a b y 7 z 0 - NAME DATE PERIOD Practice (Average) Skills Practice, p. 7 and Practice, p. 8 (shown) Polynomials Determine whether each epression is a polynomial. If it is a polynomial, state the degree of the polynomial.. y 6y yes;. ac a d m yes; 8. 8 n 9 (m n) no. z 78 yes;. 6c 6 c no 6. r s no Simplify. 7. (n ) (8n 8) 8. (6w w ) ( 7w ) n 7 8w 6w 9. (6n n ) (n 9n ) 0. (8 ) ( ) 9n n 8. (m mp 6p ) (m mp p ). ( y y ) ( y y ) 8m 7mp 7p y y. (t 7) (t t ). (u ) (6 u u) t 8t u u 0. 9( y 7w) 6. 9r y (ry 7 r y 8r 0 ) 9y 6w 7r y 9 8r 7 y 6 7r y 7. 6a w(a w aw ) 8. a w (a w 6 a w 9aw 6 ) 6a w 6a w a w 9 a 6 w a w 9 9. ( y y ) 0. ab d (ab d ab) y y a b d 7 a b d. (v 6)(v ). (7a 9y)(a y) v v a ay 9y. ( y 8). ( y) y 6y 6 0 y y. ( w)( w) 6. (n )(n ) 6w n (w s)(w ws s ) 8. ( y)( y y ) w 8s y y y 9. BANKING Terry invests $00 in two mutual funds. The first year, one fund grows.8% and the other grows 6%. Write a polynomial to represent the amount Terry s $00 grows to in that year if represents the amount he invested in the fund with the lesser growth rate GEOMETRY The area of the base of a rectangular bo measures square units. The height of the bo measures units. Find a polynomial epression for the volume of the bo. units Reading to Learn Mathematics, p. 9 - NAME 8 DATE PERIOD Reading to Learn Mathematics Polynomials Pre-Activity How can polynomials be applied to financial situations? Read the introduction to Lesson - at the top of page 9 in your tetbook. Suppose that Shenequa decides to enroll in a five-year engineering program rather than a four-year program. Using the model given in your tetbook, how could she estimate the tuition for the fifth year of her program? (Do not actually calculate, but describe the calculation that would be necessary.) Multiply $,60 by.0. Reading the Lesson. State whether the terms in each of the following pairs are like terms or unlike terms. a.,y unlike terms b. m,m like terms c. 8r,8s unlike terms d. 6, 6 like terms. State whether each of the following epressions is a monomial, binomial, trinomial, or not a polynomial. If the epression is a polynomial, give its degree. a. r r trinomial; degree b. not a polynomial c. y binomial; degree d. ab ab 6ab trinomial; degree. a. What is the FOIL method used for in algebra? to multiply binomials b. The FOIL method is an application of what property of real numbers? Distributive Property c. In the FOIL method, what do the letters F, O, I, and L mean? first, outer, inner, last d. Suppose you want to use the FOIL method to multiply ( )( ). Show the terms you would multiply, but do not actually multiply them. F ()() O ()() I ()() L ()() Helping You Remember ELL. You can remember the difference between monomials, binomials, and trinomials by thinking of common English words that begin with the same prefies. Give two words unrelated to mathematics that start with mono-, two that begin with bi-, and two that begin with tri-. Sample answer: monotonous, monogram; bicycle, bifocal; tricycle, tripod Lesson - Polynomials Lesson -

21 Assess Open-Ended Assessment Writing Have students write an eplanation, including an eample, showing why the FOIL method is a valid alternative to applying the Distributive Property when multiplying two binomials. Getting Ready for Lesson - PREREQUISITE SKILL Lesson - presents dividing polynomials. Dividing polynomials requires the use of the properties of eponents. Eercises should be used to determine your students familiarity with the properties of eponents. Answers 6. O y y 8. y y. 9c cd 7d R W RR RW R RW W WW Genetics The possible genes of parents and offspring can be summarized in a Punnett square, such as the one above. Source: Biology: The Dynamics of Life Standardized Test Practice Simplify. 7. (p 6)(p ) p p 8. (a 6)(a ) a 9a 8 9. (b )(b ) b 0. (6 z)(6 z) 6 z. ( 8)( 6) 6 8. (y 6)(y 7) 8y 6y. (a b)(a b) a 6 b. (m )(m ) m 7m. ( y) 6y 9y 6. ( c) 8c 6c 7. d (d d d ) d d 8. y (y y y ) 9. (b c) 7b 7b c 9bc c 0. ( y y )( y) y. Simplify (c 6cd d ) (7c cd 8d ) (c cd d ).. Find the product of 6 and GENETICS Suppose R and W represent two genes that a plant can inherit from its parents. The terms of the epansion of (R W) represent the possible pairings of the genes in the offspring. Write (R W) as a polynomial. R RW W. CRITICAL THINKING What is the degree of the product of a polynomial of degree 8 and a polynomial of degree 6? Include an eample in support of your answer. ; Sample answer: ( 8 )( 6 ) 8 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 8A 8B. How can polynomials be applied to financial situations? Include the following in your answer: an eplanation of how a polynomial can be applied to a situation with a fied percent rate of increase, two epressions in terms of r for the tuition in the fourth year, and an eplanation of how to use one of the epressions and the % rate of increase to estimate Shenequa s tuition in the fourth year, and a comparison of the value you found to the value given in the table. 6. Which polynomial has degree? D A C B D 7. ( y) (y z) ( z)? B A C y z y B D z y z 6. y y O 6. y y O Maintain Your Skills Mied Review Getting Ready for the Net Lesson Chapter Polynomials Simplify. Assume that no variable equals 0. (Lesson -) 8. (d ) 6d 6 9. rt (rt) 0r t 60. yz y z z y 6. ab 6a b 6. Solve the system y 0, y by using inverse matrices. (Lesson -8) (, ) Graph each inequality. (Lesson -7) 6 6. See margin. b 6. y 6. y 6. y PREREQUISITE SKILL Simplify. Assume that no variable equals 0. (To review properties of eponents, see Lesson -.) y y y 68. y y ab a y ab Differentiated Instruction Logical Have students demonstrate how to use algebra tiles to multiply two binomials that contain at least one negative coefficient. a Chapter Polynomials

22 Dividing Polynomials Lesson Notes Vocabulary synthetic division Divide polynomials using long division. Divide polynomials using synthetic division. USE LONG DIVISION In Lesson -, you learned to divide monomials. You can divide a polynomial by a monomial by using those same skills. Eample Divide a Polynomial by a Monomial Simplify y 8y y. y y 8y y y 8 y y y y Sum of quotients y y y 8 y y Eample can you use division of polynomials in manufacturing? A machinist needed square inches of metal to make a square pipe 8 inches long. In figuring the area needed, she allowed a fied amount of metal for overlap of the seam. If the width of the finished pipe will be inches, how wide is the seam? You can use a quotient of polynomials to help find the answer. s Metal Needed 8 s width of seam y y y 0 or Division Algorithm Use long division to find (z z ) (z ). z z 6 z z z z z z ()z z z(z ) z z ()z z 6z z (z) 6z 6z ()6z 0 The quotient is z 6. The remainder is 0. Divide. You can use a process similar to long division to divide a polynomial by a polynomial with more than one term. The process is known as the division algorithm. When doing the division, remember that you can only add or subtract like terms. Finished Pipe 8 Focus -Minute Check Transparency - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. 0C. can you use division of polynomials in manufacturing? Ask students: What does the epression shown in the figure represent? one half of the side length of the pipe opening What happens to the width of the pipe opening as the length of the pipe increases? The width of the pipe opening,, increases also. Teach USE LONG DIVISION In-Class Eample Power Point a Simplify b ab 0a b. ab a b a b Lesson - Dividing Polynomials Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp. Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. Enrichment, p. 6 Assessment, p. 07 Resource Manager Transparencies -Minute Check Transparency - Answer Key Transparencies Technology Interactive Chalkboard Lesson - Lesson Title

23 In-Class Eamples In-Class Eample Use long division to find ( ) ( ). Power Point Teaching Tip If students are having difficulty with the use of the division algorithm, review the algorithm as it is used for long division of numbers. Which epression is equal to (a a )( a)? D A a B a a C a a D a a USE SYNTHETIC DIVISION Power Point Teaching Tip When discussing Eample, stress that the divisor must be of the form r in order to use synthetic division. Use synthetic division to find ( 6 ) ( ). Teaching Tip Ask students to discuss whether they would rather use long division or synthetic division, giving a reason for their choice. Standardized Test Practice Test-Taking Tip You may be able to eliminate some of the answer choices by substituting the same value for t in the original epression and the answer choices and evaluating. Just as with the division of whole numbers, the division of two polynomials may result in a quotient with a remainder. Remember that 9 R and is often written as. The result of a division of polynomials with a remainder can be written in a similar manner. Eample Multiple-Choice Test Item Which epression is equal to (t t 9)( t)? A t 8 B t t 8 C t 8 D t t 8 t Eample Quotient with Remainder Read the Test Item Since the second factor has an eponent of, this is a division problem. (t t 9)( t) t t 9 t Solve the Test Item t 8 t t t9 For ease in dividing, rewrite t as t. ()t t t(t ) t t 8t 9 ()8t 0 t (t) 8t 8(t ) 8t 0 Subtract. 9 (0) The quotient is t 8, and the remainder is. Therefore, (t t 9)( t) t 8. The answer is C. t USE SYNTHETIC DIVISION Synthetic division is a simpler process for dividing a polynomial by a binomial. Suppose you want to divide 0 8 by using long division. Compare the coefficients in this division with those in Eample. Synthetic Division Use synthetic division to find ( 0 8) ( ). 0 8 () 0 0 () 6 8 () 8 0 Standardized Test Practice Step Step Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients as shown at the right. Write the constant r of the divisor r to the left. In this case, r. Bring the first coefficient,, down as shown Eample Point out that the denominator t is rewritten as t before starting the division in order to have both numerator and denominator written in descending order of the variable. Point out that the first step in the long division eliminates choice A. Students could also quickly eliminate choice B by multiplying t 8 by t and noting that the product is not t t 9. Chapter Polynomials Teacher to Teacher Christine Waddell Albion M.S., Sandy, UT "To help students better understand the division algorithm for polynomials, I first work through a long division problem with large whole numbers, such as 8, step by step. Then I work through Eample and point out the similarities in each process." Chapter Polynomials

24 Step Multiply the first coefficient by r: 0. Write the product under the second coefficient. Then add the product and the second coefficient: 0. Step Multiply the sum,, by r: () 6. Write the product under the net coefficient and add: 0 (6). Step Multiply the sum,, by r: 8. Write the product under the net coefficient and add: The remainder is 0. The numbers along the bottom row are the coefficients of the quotient. Start with the power of that is one less than the degree of the dividend. Thus, the quotient is In-Class Eample Power Point Use synthetic division to find (y y y ) (y ). y y y y Teaching Tip Remind students to include a coefficient of 0 for any missing terms of the variable in the dividend. To use synthetic division, the divisor must be of the form r. If the coefficient of in a divisor is not, you can rewrite the division epression so that you can use synthetic division. Eample Divisor with First Coefficient Other than Use synthetic division to find (8 ) ( ). Use division to rewrite the divisor so it has a first coefficient of. 8 (8 ) Divide numerator and ( ) denominator by. Simplify the numerator and denominator. Since the numerator does not have an -term, use a coefficient of 0 for. r, so r. 0 The result is. Now simplify the fraction. Rewrite as a division epression. Multiply by the reciprocal. Multiply. The solution is. (continued on the net page) Lesson - Dividing Polynomials Unlocking Misconceptions Subtracting Have students analyze any errors they make when using long division. Verify that they are using the signs correctly. Remainders Have students do a simple numeric division eample, such as 8, to help them remember how to write the remainder as part of the quotient. Lesson - Dividing Polynomials

25 Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. add the Test-Taking Tip on p. to their list of tips which they can review as they prepare for standardized tests. include any other item(s) that they find helpful in mastering the skills in this lesson. About the Eercises Organization by Objective Use Long Division: 0, 8 Use Synthetic Division:, 9, 0 Odd/Even Assignments Eercises 0 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: odd, 9,,,, 8 7 Average: odd,,, 8 7 Advanced: 6 even, 68 (optional: 69 7) All: Practice Quiz ( 0) FIND THE ERROR Suggest that students recheck their calculations immediately whenever they begin to get large numbers as the coefficients of their quotient. While nothing forbids large coefficients, this is sometimes the first indication that they have made an error in their calculations. Concept Check. The divisor contains an term.. Jorge; Shelly is subtracting in the columns instead of adding b b Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6 0,,, Standardized Test Practice indicates increased difficulty Practice and Apply 7. c d d 8. n mn m 6 Chapter Polynomials CHECK Divide using long division. 8 0 ()8 () () () The result is.. OPEN ENDED Write a quotient of two polynomials such that the remainder is. Sample answer: ( ) ( ). Eplain why synthetic division cannot be used to simplify.. FIND THE ERROR Shelly and Jorge are dividing by. Shelly Who is correct? Eplain your reasoning. Jorge Simplify. 7. a 9a 7a 6 8. z z z z 0 6y. y y y 6y. (ab ab 7a b)(ab) b 7a 6. ( 0 ) ( ) 7. (a 6a a a 6) (a ) 8. (z z 0) (z ) 9. ( y ) ( y) y y (b b b b )(b ). (y 6y ) (6y ). 9b 9b 0 b b y. Which epression is equal to ( 6)( )? B A B C D Simplify.. 9a b 8a b ab 6b y 6. 6y y y 6 y y a b y 7. (8c d cd 6cd ) (cd) 8. (mn 9m n m n) (mn) 9. (y z y z 8y z )(yz) 0. (a b a b a)(ab) y yz 8y z a b a b Differentiated Instruction Interpersonal To help discover confusions and catch careless errors, have students work in pairs as they do division problems. One person should write the solution steps, eplaining each step out loud while the other person watches, listens, and checks the work. The students should then echange roles and repeat the activity. 6 Chapter Polynomials

26 Homework Help For See Eercises Eamples 0,, 9,, 0, 8, 9 8,, Etra Practice See page See pp. 8A 8B. Cost Analyst Cost analysts study and write reports about the factors involved in the cost of production. Online Research For information about a career in cost analysis, visit: careers. $ Sample answer: r 9r 7r 8 and r. (b 8b 0b) (b ). ( ) ( ). (n n n ) (n ). (c c c ) (c ). ( ) ( ) 6. (6w 8w 0) (w ) 7. ( ) ( ) 8. ( 7) ( ) 9. y y y m 0. m 7m y m a. a a 0 m. m 0m 8 a m. 7. c c c c. (g 8g )(g ) 6. (b b b )(b ) 7. (t t 0)(t ) 8. (y )(y ) 9. (6t t 9) (t ) 0. (h h h) (h ). 9d d 8. d What is divided by? 0. Divide y y y by y. y y. BUSINESS A company estimates that it costs dollars to produce units of a product. Find an epression for the average cost per unit.. ENTERTAINMENT A magician gives these instructions to a volunteer. Choose a number and multiply it by. Then add the sum of your number and 8 to the product you found. Now divide by the sum of your number and. What number will the volunteer always have at the end? Eplain. ; See margin for eplanation. MEDICINE For Eercises and, use the following information. The number of students at a large high school who will catch the flu during 70t an outbreak can be estimated by n t, where t is the number of weeks from the beginning of the epidemic and n is the number of ill people. 70t. Perform the division indicated by 70 t. 70 t. Use the formula to estimate how many people will become ill during the first week. 8 people PHYSICS For Eercises 7, suppose an object moves in a straight line so that after t seconds, it is t t 6t feet from its starting point.. 6 ft Lesson - Dividing Polynomials 7 Answer. Find the distance the object travels between the times t and t. 6. How much time elapses between t and t? s 7. Find a simplified epression for the average speed of the object between times t and t. ft/s 8. CRITICAL THINKING Suppose the result of dividing one polynomial by another is r 6r 9. What two polynomials might have been divided? r. Let be the number. Multiplying by results in. The sum of the number, 8, and the result of the multiplication is 8 or 8. Dividing by the sum 8 of the number and gives or. The end result is always. NAME DATE PERIOD - Enrichment, p. 6 Oblique Asymptotes The graph of y a b, where a 0, is called an oblique asymptote of y f() if the graph of f comes closer and closer to the line as or. is the mathematical symbol for infinity, which means endless. For f(), y is an oblique asymptote because f(), and 0 as or. In other words, as increases, the value of gets smaller and smaller approaching 0. Eample Find the oblique asymptote for f() 8. 8 Use synthetic division. 6 y 8 6 Study Study Guide and and Intervention Intervention, p. (shown) and p. - NAME DATE PERIOD Dividing Polynomials Use Long Division To divide a polynomial by a monomial, use the properties of powers from Lesson -. To divide a polynomial by a polynomial, use a long division pattern. Remember that only like terms can be added or subtracted. Eample p t r p qtr 9p tr p tr p t r p qtr 9p tr Simplify. p tr p t r p qtr 9p tr p tr p tr p tr 9 p t r p qt r p t r pt 7qr p Eample Use long division to find ( 8 9) ( ). 8 9 () () 6 9 () 8 7 The quotient is, and the remainder is 7. Therefore Eercises Simplify. 8a. 0a mn. 6 0m n 60a. b 8b 8a b a m n ab 6a 6n b 0a 0 ab 7a m a. ( ) ( ). (m m 7) (m ) m m 6. (p 6) (p ) 7. (t 6t ) (t ) p p p t 8t 6 t 8. ( ) ( ) 9. ( ) ( ) - NAME DATE PERIOD Practice (Average) Skills Practice, p. and Practice, p. (shown) Dividing Polynomials Simplify. r. 0 r 8 0r r 6 r 8 6k k. m k m 9m 6k 9m r r km m k. (0 y y 8 y) (6 y). (6w z w z w z) (w z) y wz z wz w. (a 8a a )(a) 6. (8d k d k dk )(dk ) a a a 7d d f 7. 7f 0 f 8. f 7 9. (a 6) (a ) a a 6 0. (b 7) (b ) b b (w 7w w ) (w ). (6y y 8y 6) (y ) w w 6y w y 6 6y 6 y. ( 0) ( ) 6. (m m ) (m ) m m m m m 7. ( 6)( ) 8. (6y y )(y ) 6 0 y 7 y (r r r ) (r ). (6t t t ) (t ) r r t t t p. 7p p h. h h h p h p p p h h. GEOMETRY The area of a rectangle is square feet. The length of the rectangle is feet. What is the width of the rectangle? ft 6. GEOMETRY The area of a triangle is square meters. The length of the base of the triangle is 6 meters. What is the height of the triangle? m Reading to Learn Mathematics, p. - NAME DATE PERIOD Reading to Learn Mathematics Dividing Polynomials Pre-Activity How can you use division of polynomials in manufacturing? Read the introduction to Lesson - at the top of page in your tetbook. Using the division symbol (), write the division problem that you would use to answer the question asked in the introduction. (Do not actually divide.) ( ) (8) Reading the Lesson. a. Eplain in words how to divide a polynomial by a monomial. Divide each term of the polynomial by the monomial. b. If you divide a trinomial by a monomial and get a polynomial, what kind of polynomial will the quotient be? trinomial. Look at the following division eample that uses the division algorithm for polynomials Which of the following is the correct way to write the quotient? C A. B. C. D.. If you use synthetic division to divide 8 by, the division will look like this: Which of the following is the answer for this division problem? B A. B. C. D. Helping You Remember ELL. When you translate the numbers in the last row of a synthetic division into the quotient and remainder, what is an easy way to remember which eponents to use in writing the terms of the quotient? Sample answer: Start with the power that is one less than the degree of the dividend. Decrease the power by one for each term after the first. The final number will be the remainder. Drop any term that is represented by a 0. Lesson - Lesson - Lesson - Dividing Polynomials 7

27 Assess Open-Ended Assessment Writing Have students write their own list of tips for how to do division problems, describing the techniques they use to help avoid making errors. Intervention Some students may have trouble keeping their concentration throughout the sequence of steps required in long division. Encourage them to compare intermediate results with a partner, so that they can ask questions and catch errors before completing the entire problem. New Getting Ready for Lesson - BASIC SKILL Lesson - presents factoring polynomials. This requires a knowledge of the greatest common factor. Eercises 69 7 should be used to determine your students familiarity with the greatest common factor of a set of numbers. Assessment Options Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons - through -. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not yet mastered. Quiz (Lessons - through -) is available on p. 07 of the Chapter Resource Masters. P Standardized Test Practice Maintain Your Skills Mied Review Getting Ready for the Net Lesson ractice Quiz Epress each number in scientific notation. (Lesson -). 6,000, Simplify. (Lessons - and -). ( y) () 08 8 y. a 6bc a. ab c b c z z z 6 6. (9 y) (7 y) y 7. (t )(t ) t t 8 8. (n )(n n ) Simplify. (Lesson -) 9. m 9 n n n m 9. (m m m 7) (m ) 0. d d 9d 9 d d d 8 Chapter Polynomials 9. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 8A 8B. How can you use division of polynomials in manufacturing? Include the following in your answer: the dimensions of the piece of metal that the machinist needs, the formula from geometry that applies to this situation, and an eplanation of how to use division of polynomials to find the width s of the seam. 60. An office employs women and men. What is the ratio of the total number of employees to the number of women? A A B C D 6. If a b c and a b, then all of the following are true EXCEPT D A C a c b c. B a b 0. a b c. D c b a. Simplify. (Lesson -) y z y z y z 6. ( ) ( 9) 6. y z(y z yz ) 6. (y )(y ) y y 6. (a b) a ab b 66. ASTRONOMY Earth is an average of. 0 meters from the Sun. Light travels at 0 8 meters per second. About how long does it take sunlight to reach Earth? (Lesson -) 0 s or 8 min 0 s Write an equation in slope-intercept form for each graph. (Lesson -) 67. y y 68. y y O (, ) (, ) (, ) BASIC SKILL Find the greatest common factor of each set of numbers , , , 8 7., 7, 8 7., 0, 6 7., 0, 6 O (, 0) Lessons - through - 8 Chapter Polynomials

28 Factoring Polynomials Lesson Notes Factor polynomials. Simplify polynomial quotients by factoring. does factoring apply to geometry? Suppose the epression 0 6 represents the area of a rectangle. Factoring can be used to find possible dimensions of the rectangle.? units A 0 6 units? units Focus -Minute Check Transparency - Use as a quiz or review of Lesson -. FACTOR POLYNOMIALS Whole numbers are factored using prime numbers. For eample, 00. Many polynomials can also be factored. Their factors, however, are other polynomials. Polynomials that cannot be factored are called prime. The table below summarizes the most common factoring techniques used with polynomials. Number of Terms Factoring Technique General Case Factoring Techniques any number Greatest Common Factor (GCF) a b a b ab ab(a b a b) two Difference of Two Squares a b (a b)(a b) Sum of Two Cubes a b (a b)(a ab b ) Difference of Two Cubes a b (a b)(a ab b ) three Perfect Square Trinomials a ab b (a b) a ab b (a b) General Trinomials ac (ad bc) bd (a b)(c d) four or more Grouping a b ay by (a b) y(a b) (a b)( y) Whenever you factor a polynomial, always look for a common factor first. Then determine whether the resulting polynomial factor can be factored again using one or more of the methods listed in the table above. Eample GCF Factor 6 y y 6 y. 6 y y 6 y ( y y) ( y y) ( y) Check this result by finding the product. (y y) (y y) (y ) y(y y ) The GCF is y. The remaining polynomial cannot be factored using the methods above. Mathematical Background notes are available for this lesson on p. 0D. Building on Prior Knowledge In this lesson, students will need to recall how to find the area of a rectangle, and they will also need to remember the set of prime numbers. does factoring apply to geometry? Ask students: How do Eamples and in Lesson - relate to factoring, the topic of this lesson? The quotient and the divisor are factors of the dividend. Teach FACTOR POLYNOMIALS In-Class Eample Power Point Factor 0a b a b ab. ab(a b a b ) A GCF is also used in grouping to factor a polynomial of four or more terms. Lesson - Factoring Polynomials 9 Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp. 7 8 Skills Practice, p. 9 Practice, p. 60 Reading to Learn Mathematics, p. 6 Enrichment, p. 6 Graphing Calculator and Spreadsheet Masters, p. Teaching Algebra With Manipulatives Masters, p. Transparencies -Minute Check Transparency - Answer Key Transparencies Technology AlgePASS: Tutorial Plus, Lesson 8 Interactive Chalkboard Lesson - Lesson Title 9

29 In-Class Eample Power Point Factor 0. ( )( ) Teaching Tip Point out to students that it is often difficult to recognize that grouping can be used to factor a polynomial. Stress that this technique should only be considered when trying to factor a polynomial with four terms. Study Tip Algebra Tiles When modeling a polynomial with algebra tiles, it is easiest to arrange the tiles first, then the tiles and finally the tiles to form a rectangle. Eample Grouping Factor a a a. a a a (a a ) (a ) a (a ) (a ) (a )(a ) Factoring Trinomials Use algebra tiles to factor 7. Model and Analyze Use algebra tiles to model 7. To find the product that resulted in this polynomial, arrange the tiles to form a rectangle. Notice that the total area can be epressed as the sum of the areas of two smaller rectangles. Group to find a GCF. Factor the GCF of each binomial. Distributive Property You can use algebra tiles to model factoring a polynomial. 6 Use these epressions to rewrite the trinomial. Then factor. 7 ( ) (6 ) total area sum of areas of smaller rectangles ( ) ( ) Factor out each GCF. ( )( ) Distributive Property. 6; It is the same.. Find two numbers with a product of or 6 and a sum of 7. Use those numbers to rewrite the trinomial. Then factor. Make a Conjecture Study the factorization of 7 above.. What are the coefficients of the two terms in ( ) (6 )? Find their sum and their product. and 6; 7; 6. Compare the sum you found in Eercise to the coefficient of the term in 7. They are the same.. Find the product of the coefficient of the term and the constant term in 7. How does it compare to the product in Eercise?. Make a conjecture about how to factor 7. The FOIL method can help you factor a polynomial into the product of two binomials. Study the following eample. F O I L (a b)(c d) a c a d b c b d ac (ad bc) bd Notice that the product of the coefficient of and the constant term is abcd. The product of the two terms in the coefficient of is also abcd. 0 Chapter Polynomials Algebra Activity Materials: algebra tiles Inform students that the two factors of the trinomial can be read directly from the completed array of tiles in another way. Point out that the length of the array is, and the height is. The product of the length and width gives the area, 7, of the array. You might wish to have students eperiment to see if there is another way to form a rectangle with the tiles. 0 Chapter Polynomials

30 Eample Two or Three Terms Factor each polynomial. a. 6 To find the coefficients of the -terms, you must find two numbers whose product is 6 or 0, and whose sum is. The two coefficients must be 0 and since (0)() 0 and 0 (). Rewrite the epression using 0 and in place of and factor by grouping Substitute 0 for. ( 0) ( 6) Associative Property ( ) ( ) ( )( ) b. y 8 y 8 (y 6) (y )(y ) You can use a graphing calculator to check that the factored form of a polynomial is correct. Factoring Polynomials Factor out the GCF. Is the factored form of equal to ( 7)( )? You can find out by graphing y and y ( 7)( ). If the two graphs coincide, the factored form is probably correct. Enter y and y ( 7)( ) on the Y= screen. Graph the functions. Since two different graphs appear, ( 7)( ). Factor out the GCF of each group. Distributive Property y 6 is the difference of two squares. c. c d 7 c d (cd) and 7. Thus, this is the sum of two cubes. c d 7 (cd )[(cd) (cd) ] Sum of two cubes formula with a cd and b d. m 6 n 6 (cd )(c d cd 9) Simplify. This polynomial could be considered the difference of two squares or the difference of two cubes. The difference of two squares should always be done before the difference of two cubes. This will make the net step of the factorization easier. m 6 n 6 (m n )(m n ) Difference of two squares (m n)(m mn n )(m n)(m mn n ) Sum and difference of two cubes [0, 0] scl: by [0, 0] scl: In-Class Eample Power Point Factor each polynomial. a. y y (y )(y ) b. mp m m(p )(p ) c. y 8 (y )( y y ) d. 6 6 y 6 ( y)( y y )( y)( y y ) Teaching Tip Emphasize the importance of checking each factor to make sure it is prime before deciding that the final group of factors has been found. Concept Check Ask students to write an ordered list describing what they will check for as they factor a polynomial. Sample answer: First look for any common factors of the terms; if there is a common factor, find the GCF. After factoring out the GCF, look for the difference of two squares, a perfect square trinomial, and so on. Use the factoring techniques listed on p. 9 to factor the epression further. Eamine the resulting factored form to see if there are any factors that are not prime. If so, continue the process. If not, the factoring is complete.. No; in some cases, the graphs might be so close in shape that they seem to coincide but do not. Think and Discuss. Determine if 6 ( )( ) is a true statement. If not, write the correct factorization. no; ( 6)( ). Does this method guarantee a way to check the factored form of a polynomial? Why or why not? Lesson - Factoring Polynomials Factoring Polynomials So that students see what happens when a polynomial and a correct factorization are graphed, have students graph the functions y 8 and y ( 9)( 9) in the same screen. It looks like only one graph appears on the screen because both graphs are the same. Lesson - Factoring Polynomials

31 SIMPLIFY QUOTIENTS In-Class Eample a Simplify a 6. a 7a 0 a, if a, a Power Point Practice/Apply SIMPLIFY QUOTIENTS In Lesson -, you learned to simplify the quotient of two polynomials by using long division or synthetic division. Some quotients can be simplified using factoring. Eample Quotient of Two Trinomials Simplify. 7 ( ) ( ) Factor the numerator and denominator. 7 ( ) ( ) Divide. Assume,. Therefore,, if,. 7 Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. add a list of factoring techniques to their notebook, including the factoring of the special cases listed in the Concept Summary on p. 9 and the FOIL method described on p. 0. include any other item(s) that they find helpful in mastering the skills in this lesson. Concept Check. Sample answer: Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7, Application. OPEN ENDED Write an eample of a perfect square trinomial.. Find a countereample to the statement a b (a b).. Decide whether the statement is sometimes, always, or 6 never true. sometimes. Sample answer: If a and b, then a b but (a b). Factor completely. If the polynomial is not factorable, write prime.. 6 6( ). a a ab a(a b) 6. 7y y ( 7)( y) 7. y 6y 8 (y )(y ) 8. z z (z 6)(z ) 9. b 8 (b )(b ) 0. 6w 69 (w )(w ). h 8000 (h 0)(h 0h 00) Simplify. Assume that no denominator is equal to y 8y y 7 y 6 y. GEOMETRY Find the width of rectangle ABCD if its area is 9y 6y square centimeters. y cm A 6y cm D About the Eercises Organization by Objective Factor Polynomials: Simplify Quotients: 6 Odd/Even Assignments Eercises and 6 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 7 odd, 9 odd, 8, 6 8 Average: odd, 8, 6 8 (optional: 9 6) Advanced: 6 0 even, 7 (optional: 76 8) indicates increased difficulty Practice and Apply 7. cd (6d c c d) Chapter Polynomials Factor completely. If the polynomial is not factorable, write prime.. y 0 (y ) 6. 6a b 8ab 6ab (a b) 7. cd 8c d 0c d 8. a b c y ad y prime 9. 8yz 6z y 9 (z )(y ) 0. a a (a )( ). 7 6 ( )( 6). y y (y )(y ). a a (a )(a ). b b 7 (b )(b 7). 6c c 6 (c )(c ) 6. m m 6 (m )(m ) 7. n n (n 8)(n ) 8. z z (z )(z ) Differentiated Instruction Auditory/Musical Ask students to create songs or raps to help them remember the factoring techniques for the difference of two squares, for the sum or difference of two cubes, or for one of the two perfect square trinomial types. B C Chapter Polynomials

32 Homework Help For Eercises See Eamples 8 9, 0 8,, 9, 6 Etra Practice See page ( 6) ( ). 6a b prime. m n (m n)(m n). y z (y z)(y z). 7y ( y)( y). z (z )(z z ) 6. t 8 (t )(t t ) 7. p (p )(p )(p ) 8. 8 ( 9)( )( ) 9. 7ac bc 7ad bd (7a b)(c d)(c d) y 8z y z (8 )( y z). a aby acz ab b y bcz (a b)(a by cz). a a a 9a b 6ab b (a b)(a )(a ) Study Study Guide and and Intervention Intervention, p. 7 (shown) and p. 8 - NAME DATE PERIOD Factoring Polynomials Factor Polynomials Techniques for Factoring Polynomials For any number of terms, check for: greatest common factor For two terms, check for: Difference of two squares a b (a b)(a b) Sum of two cubes a b (a b)(a ab b ) Difference of two cubes a b (a b)(a ab b ) For three terms, check for: Perfect square trinomials a ab b (a b) a ab b (a b) General trinomials ac (ad bc) bd (a b)(c d) For four terms, check for: Grouping a b ay by (a b) y(a b) (a b)( y). Find the factorization of. ( )( ). What are the factors of y 9y? (y )(y ). LANDSCAPING A boardwalk that is feet wide is built around a rectangular pond. The combined area of the pond and the boardwalk is 0 00 square feet. What are the dimensions of the pond? 0 ft by 0 ft a Eample Factor. First factor out the GCF to get (8 ). To find the coefficients of the terms, you must find two numbers whose product is 8 () 0 and whose sum is. The two coefficients must be 0 and 6. Rewrite the epression using 0 and 6 and factor by grouping Group to find a GCF. ( ) ( ) Factor the GCF of each binomial. ( )( ) Distributive Property Thus, ( )( ). Eercises Factor completely. If the polynomial is not factorable, write prime.. y y. 6mn 8m n. 8 6 y ( y) (6m )(n ) ( 8)( ).. y 60 y 6. r 0 ( )( )( ) y(7y ) (r )(r r ) Lesson m c c c c (0m )(0m ) prime c(c ) (c ) Buildings The tallest buildings in the world are the Petronas Towers in Kuala Lumpur, Malaysia. Each is 8 feet tall. Source: Simplify. Assume that no denominator is equal to ( ) ( ). ( )( ) BUILDINGS For Eercises and, use the following information. When an object is dropped from a tall building, the distance it falls between second after it is dropped and seconds after it is dropped is 6 6 feet.. How much time elapses between second after it is dropped and seconds after it is dropped? s. What is the average speed of the object during that time period? 6 6 ft/s GEOMETRY The length of one leg of a right triangle is 6 centimeters, and the area is 7 square centimeters. What is the length of the other leg? 8 cm. CRITICAL THINKING Factor 6p n 6p n. (8p n ) 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 8A 8B. How does factoring apply to geometry? Include the following in your answer: an eplanation of how to use factoring to find possible dimensions for the rectangle described at the beginning of the lesson, and why your dimensions are not the only ones possible, even if you assume that the dimensions are binomials with integer coefficients. Lesson - Factoring Polynomials - NAME 7 DATE PERIOD Practice (Average) Skills Practice, p. 9 and Practice, p. 60 (shown) Factoring Polynomials Factor completely. If the polynomial is not factorable, write prime.. a b 0ab. st 9s t 6s t. y y y ab(a b) st(t s st) y( y ). y y y y. 7t r rt 6. y y y( y y ) (7 r)( t) ( )( y) 7. y 0y ab a 6b 9. 6n n (y 8)(y ) (a )(b ) (6n )(n ) p 7p ( )( ) prime (p 9)(p ). r r r. 8a a 6. c 9 r(r 9)(r 6) (a )(a ) (c 7)(c 7) r b 8 ( )( ) (r )(r ) (b 9)(b )(b ) 9. 8m prime 0. t t 8t t(t 8). y y y (y )(y y 9). 8 6 (9 )( )( ) Simplify. Assume that no denominator is equal to DESIGN Bobbi Jo is using a software package to create a drawing of a cross section of a brace as shown at the right. Write a simplified, factored epression that represents the area of the cross section of the brace. (0. ) cm COMBUSTION ENGINES In an internal combustion engine, the up and down motion of the pistons is converted into the rotary motion of the crankshaft, which drives the flywheel. Let r represent the radius of the flywheel at the right and let r represent the radius of the crankshaft passing through it. If the formula for the area of a circle is A r, write a simplified, factored epression for the area of the cross section of the flywheel outside the crankshaft. (r r )(r r ) Reading to Learn Mathematics, p. 6 - NAME 60 DATE PERIOD Reading to Learn Mathematics Factoring Polynomials cm ( ) 9 Pre-Activity How does factoring apply to geometry? Read the introduction to Lesson - at the top of page 9 in your tetbook. If a trinomial that represents the area of a rectangle is factored into two binomials, what might the two binomials represent? the length and width of the rectangle Reading the Lesson. Name three types of binomials that it is always possible to factor. difference of two squares, sum of two cubes, difference of two cubes. Name a type of trinomial that it is always possible to factor. perfect square trinomial. Complete: Since y cannot be factored, it is an eample of a polynomial. cm prime 8. cm r ELL r cm NAME DATE PERIOD - Enrichment, p. 6 Using Patterns to Factor Study the patterns below for factoring the sum and the difference of cubes. a b (a b)(a ab b ) a b (a b)(a ab b ) This pattern can be etended to other odd powers. Study these eamples. Eample Factor a b. Etend the first pattern to obtain a b (a b)(a a b a b ab b ). Check: (a b)(a a b a b ab b ) a a b a b a b ab a b a b a b ab b a b Eample Factor a b. Etend the second pattern to obtain a b (a b)(a a b a b ab b ). Check: (a b)(a a b a b ab b ) a a b a b a b ab a b ab b. On an algebra quiz, Marlene needed to factor 70. She wrote the following answer: ( )( ). When she got her quiz back, Marlene found that she did not get full credit for her answer. She thought she should have gotten full credit because she checked her work by multiplication and showed that ( )( ) 70. a. If you were Marlene s teacher, how would you eplain to her that her answer was not entirely correct? Sample answer: When you are asked to factor a polynomial, you must factor it completely. The factorization was not complete, because can be factored further as ( 7). b. What advice could Marlene s teacher give her to avoid making the same kind of error in factoring in the future? Sample answer: Always look for a common factor first. If there is a common factor, factor it out first, and then see if you can factor further. Helping You Remember. Some students have trouble remembering the correct signs in the formulas for the sum and difference of two cubes. What is an easy way to remember the correct signs? Sample answer: In the binomial factor, the operation sign is the same as in the epression that is being factored. In the trinomial factor, the operation sign before the middle term is the opposite of the sign in the epression that is being factored. The sign before the last term is always a plus. Lesson - Lesson - Factoring Polynomials

33 Assess Open-Ended Assessment Modeling Have students use algebra tiles to model and determine the factored form of the polynomial 6. ( )( ) Intervention Point out that factoring involving a b, a b, a b, (a b), and (a b) occurs so frequently in algebra that students should memorize these forms. Then students can easily recognize and use them whenever the need arises. New Getting Ready for Lesson - PREREQUISITE SKILL Lesson - discusses radicals. Many radicals are irrational numbers whose approimate value can be found using a calculator. Eercises 76 8 should be used to determine your students familiarity with rational and irrational numbers. Standardized Test Practice Graphing Calculator Maintain Your Skills Mied Review 7. Which of the following is the factorization of? B A C ( )( ) B ( )( ) ( )( ) D ( )( ) 8. Which is not a factor of? C A B C D CHECK FACTORING Use a graphing calculator to determine if each polynomial is factored correctly. Write yes or no. If the polynomial is not factored correctly, find the correct factorization. 60. no; ( )( ) 9. ( )( ) yes ( )( ) 6. ( )( ) 6. 8 ( )( ) yes no; ( )( ) Simplify. (Lesson -) 6. (t t ) (t ) t t 6. (y y )(y ) y Simplify. (Lesson -) 67. ( y y ) ( y y ) 68. ( )(7 ) 6 y y Perform the indicated operations, if possible. (Lesson -) [ ] [] PHOTOGRAPHY The perimeter of a rectangular picture is 86 inches. Twice the width eceeds the length by inches. What are the dimensions of the picture? (Lesson -) in. by 8 in. Determine whether each relation is a function. Write yes or no. (Lesson -) 7. y yes 7. y no O 8 0 O 8 Getting Ready for the Net Lesson State the property illustrated by each equation. (Lesson -) 7. ( 8) () 8() Distributive 7. (7 ) ( 7) Associative () PREREQUISITE SKILL Determine whether each number is rational or irrational. (To review rational and irrational numbers, see Lesson -.) 80. irrational rational 77. irrational rational rational rational Chapter Polynomials Unlocking Misconceptions Difference of Two Squares Many people think that the epressions a b and (a b) are the same. Have students choose values for a and b, such as a and b, to see that this is not true. Sum of Two Squares Students may need to be convinced that a b cannot be factored after seeing that a b can be factored. Have them substitute values for a and b to test possible factored forms, such as (a b)(a b), to verify they do not equal a b. Chapter Polynomials

34 Roots of Real Numbers Lesson Notes Vocabulary square root nth root principal root Simplify radicals. Use a calculator to approimate radicals. do square roots apply to oceanography? The speed s in knots of a wave can be estimated using the formula s., where is the length of the wave in feet. This is an eample of an equation that contains a square root. SIMPLIFY RADICALS Finding the square root of a number and squaring a number are inverse operations. To find the square root of a number n, you must find a number whose square is n. For eample, 7 is a square root of 9 since 7 9. Since (7) 9, 7 is also a square root of 9. Definition of Square Root Words For any real numbers a and b, if a b, then a is a square root of b. Eample Since, is a square root of. Since finding the square root of a number and squaring a number are inverse operations, it makes sense that the inverse of raising a number to the nth power is finding the nth root of a number. The table below shows the relationship between raising a number to a power and taking that root of a number. Powers Factors Roots a is a cube root of. a 8 8 is a fourth root of 8. a is a fifth root of. a n b a a a a a b a is an nth root of b. n factors of a Focus -Minute Check Transparency - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. 0D. do square roots apply to oceanography? Ask students: One knot means one nautical mile per hour and one nautical mile is about 6076 feet. One mile on land (called a statute mile) is 80 feet. Which is faster, knot or statute mile per hour? knot As the length of a wave (represented by in the diagram) increases, does the speed of the wave increase or decrease? increases This pattern suggests the following formal definition of an nth root. Definition of nth Root Words For any real numbers a and b, and any positive integer n, if a n b, then a is an nth root of b. Eample Since, is a fifth root of. Lesson - Roots of Real Numbers Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp. 6 6 Skills Practice, p. 6 Practice, p. 66 Reading to Learn Mathematics, p. 67 Enrichment, p. 68 Assessment, pp. 07, 09 Teaching Algebra With Manipulatives Masters, pp. 6 7 Transparencies -Minute Check Transparency - Answer Key Transparencies Technology Interactive Chalkboard Lesson - Lesson Title

35 Teach SIMPLIFY RADICALS In-Class Eample Simplify. a. 6 6 b. (q ) (q ) c. a b 0 a b d. not a real number Power Point Teaching Tip Be sure students understand that since 9 and () 9, then the equation 9 has two roots, and. However, the value of the epression 9 is only. To indicate both square roots and not just the principal root, the epression must be given as 9. Teaching Tip When discussing the information following Eample, offer this alternative. Another way to simplify radicals that involve only numbers and no variables, is to simplifying the epression under the radical sign first. For eample, () could be rewritten by first simplifying under the radical sign to get, and then taking the principal root to get. Similarly, () 6 simplifies to 6, whose principal square root is 8. Reading Tip Make sure that students understand what it means to say that the radical sign designates the principal root. New Study Tip Reading Math n 0 is read the nth root of 0. The symbol n indicates an nth root. Some numbers have more than one real nth root. For eample, 6 has two square roots, 6 and 6. When there is more than one real root, the nonnegative root is called the principal root. When no inde is given, as in 6, the radical sign indicates the principal square root. The symbol n b stands for the principal nth root of b. If n is odd and b is negative, there will be no nonnegative root. In this case, the principal root is negative. 6 6 indicates the principal square root of indicates the opposite of the principal square root of indicates both square roots of 6. means positive or negative. indicates the principal cube root of. 8 8 indicates the opposite of the principal fourth root of 8. The chart below gives a summary of the real nth roots of a number b. Eample Simplify. a. Find Roots ( ) Real nth roots of b, n b, or n b b. (y ) 8 (y ) 8 [(y ) ] (y ) The square roots of The opposite of the principal square are. root of (y ) 8 is (y ). c. y 0 y 0 ( ) y y inde The principal fifth root of y 0 is y. n d. 9 radical sign radicand n n b if b 0 n b if b 0 b 0 even odd one positive root, one negative root 6 one positive root, no negative roots 9 9 n is even. b is negative. Thus, 9 is not a real number. When you find the nth root of an even power and the result is an odd power, you must take the absolute value of the result to ensure that the answer is nonnegative. () or () 6 () or 8 If the result is an even power or you find the nth root of an odd power, there is no need to take the absolute value. Why? 0 no real roots is not a real number. no positive roots, one negative root 8 one real root, 0 n Chapter Polynomials Unlocking Misconceptions Variables Some students tend to think that must represent a positive number and must represent a negative number. Reading as the opposite of should help them understand that is if. Square Roots of Negative Numbers Eplain that 9 has no square root that is a real number. That is, no real number can be squared to give 9. However, inform students that 9 does represent a number, called an imaginary number. Lesson -9 discusses such numbers. 6 Chapter Polynomials

36 Study Tip Graphing Calculators To find a root of inde greater than, first type the inde. Then select 0 from the MATH menu. Finally, enter the radicand. Concept Check Eample Simplify. a. 8 8 APPROXIMATE RADICALS WITH A CALCULATOR Recall that real numbers that cannot be epressed as terminating or repeating decimals are irrational numbers. and are eamples of irrational numbers. Decimal approimations for irrational numbers are often used in applications. Eample PHYSICS Approimate a Square Root The time T in seconds that it takes a pendulum to make a complete swing back and forth is given by the formula T, L g where L is the length of the pendulum in feet and g is the acceleration due to gravity, feet per second squared. Find the value of T for a -foot-long pendulum in a grandfather clock. Eplore You are given the values of L and g and must find the value of T. Since the units on g are feet per second squared, the units on the time T should be seconds. Plan Simplify Using Absolute Value Substitute the values for L and g into the formula. Use a calculator to evaluate. Solve T L g Original formula L, g.9 Use a calculator. b. 8(a ) Note that is an eighth root of 8. 8(a ) [(a ) ] The inde is even, so the principal root is nonnegative. Since could Since the inde is even and the be negative, you must take the eponent is odd, you must use absolute value of to identify the the absolute value of (a ). principal root. 8(a ) (a ) 8 8 It takes the pendulum about.9 seconds to make a complete swing. Eamine The closest square to is, and is approimately, so the answer 9 should be close to () 9 () or. The answer is reasonable.. OPEN ENDED Write a number whose principal square root and cube root are both integers. Sample answer: 6. Eplain why it is not always necessary to take the absolute value of a result to indicate the principal root. See margin.. Determine whether the statement () is sometimes, always, or never true. Eplain your reasoning. Sometimes; it is true when 0. Differentiated Instruction Lesson - Roots of Real Numbers 7 Visual/Spatial Have students create various rectangles using inde cards or cardboard, or using masking tape on the classroom floor. Have them use the formula d w to find the length of the diagonal of each rectangle. After creating several rectangles, have students eperiment with using the diagonal measures of two rectangles to create another rectangle whose length and width are irrational numbers. Students should then find the length of the diagonal of this new rectangle. In-Class Eample Simplify. a. 6 t 6 t b. ( ) ( ) In-Class Eample Practice/Apply Study Notebook Power Point APPROXIMATE RADICALS WITH A CALCULATOR Power Point PHYSICS Use the formula given in Eample. Find the value of T for a.-foot-long pendulum. about.6 s Teaching Tip Help students see how time and the length of a pendulum are related by having them eperiment using weights with different lengths of string. Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. keep a list of study tips for the graphing calculator, including the one in this lesson. include any other item(s) that they find helpful in mastering the skills in this lesson. Answer. If all of the powers in the result of an even root have even eponents, the result is nonnegative without taking absolute value. Lesson - Roots of Real Numbers 7

37 Study Study Guide and and Intervention Intervention, p. 6 (shown) and p. 6 - NAME DATE PERIOD Simplify Radicals Roots of Real Numbers Square Root For any real numbers a and b, if a b, then a is a square root of b. For any real numbers a and b, and any positive integer n, if a nth Root n b, then a is an nth root of b.. If n is even and b 0, then b has one positive root and one negative root. Real nth Roots of b,. If n is odd and b 0, then b has one positive root. n b, b n. If n is even and b 0, then b has no real roots.. If n is odd and b 0, then b has one negative root. Eample Simplify 9z. 8 Eample 9z 8 (7z ) 7z z must be positive, so there is no need to take the absolute value. Eercises Simplify (a ) 6 (a ) 6 [(a ) ] (a ) Simplify p p. a 0. p 0 6. m 6 n 9 a p m n 7. b 8. 6a b b a b 0. (k). 69r. 7p 6 6k r p. 6y z. 6q. 00 y z 6 y z 6 q 7 0 y z p 0 0. not a real number 0.8 p 9. () 8 0. (y ). (a b) y a b. ( ). (m ) 6. 6 (m ) 6 NAME 6 DATE PERIOD - Skills Practice Practice, p. 6 and (Average) Practice, Roots of Real p. Numbers 66 (shown) Use a calculator to approimate each value to three decimal places (0.9) Simplify (a) 0. (a ) not a real number a. 9m t 8 6m.. 6r w 6. () 8 7 m t m r w 6. 6s p q y y 9 s 6pq 6 y y 9. m 8 n 6 0. y 0. 6 (m ) 6. ( ) m n y m. 9a b 0 6. ( ) 8. d a b 8 ( ) 7d 7. RADIANT TEMPERATURE Thermal sensors measure an object s radiant temperature, which is the amount of energy radiated by the object. The internal temperature of an object is called its kinetic temperature. The formula T r T k e relates an object s radiant temperature T r to its kinetic temperature T k. The variable e in the formula is a measure of how well the object radiates energy. If an object s kinetic temperature is 0 C and e 0.9, what is the object s radiant temperature to the nearest tenth of a degree? 9.C 8. HERO S FORMULA Salvatore is buying fertilizer for his triangular garden. He knows the lengths of all three sides, so he is using Hero s formula to find the area. Hero s formula states that the area of a triangle is s(s a)(s b)(s, c) where a, b, and c are the lengths of the sides of the triangle and s is half the perimeter of the triangle. If the lengths of the sides of Salvatore s garden are feet, 7 feet, and 0 feet, what is the area of the garden? Round your answer to the nearest whole number. ft Reading to Learn Mathematics, p NAME 66 DATE PERIOD Reading to Learn Mathematics Roots of Real Numbers Pre-Activity How do square roots apply to oceanography? Read the introduction to Lesson - at the top of page in your tetbook. Suppose the length of a wave is feet. Eplain how you would estimate the speed of the wave to the nearest tenth of a knot using a calculator. (Do not actually calculate the speed.) Sample answer: Using a calculator, find the positive square root of. Multiply this number by.. Then round the answer to the nearest tenth. Reading the Lesson. For each radical below, identify the radicand and the inde. a. radicand: inde: b. radicand: inde: c. radicand: inde: ELL Lesson - Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7, Application indicates increased difficulty Practice and Apply Homework Help For Eercises See Eamples 6 7, , Etra Practice See page not a real number. 8 Chapter Polynomials Use a calculator to approimate each value to three decimal places Simplify. 0. not a real number. 6 a b. y () y y. 6a b. ( y). OPTICS The distance D in miles from an observer to the horizon over flat land or water can be estimated using the formula D.h, where h is the height in feet of the point of observation. How far is the horizon for a person whose eyes are 6 feet above the ground? about.0 mi Use a calculator to approimate each value to three decimal places , (7) (00) 9.6 Simplify (7). (8) z 8 z m 6 7 m. 6a 8 8a. 7r r. c 6 c. (g) g. (z) 6 z 6. y 6 y 7. 6 z 6 z y y 9. 9p q 6 p 6 q 0. 8a b ab. 7c d 9 c d. ( y) y. (p ) q p q.. zz a a 7. 9 z a not a real number 8. Find the principal fifth root of. 9. What is the third root of? 60. SPORTS Refer to the drawing at the right. How far does the catcher have to throw a ball from home plate to second base? about 7.8 ft 6. FISH The relationship between the length and mass of Pacific halibut can be approimated by the equation L 0.6 M, where L is the length in meters and M is the mass in kilograms. Use this equation to predict the length of a -kilogram Pacific halibut. about. m rd base 90 ft 90 ft nd base pitcher home plate catcher 90 ft 90 ft st base. Complete the following table. (Do not actually find any of the indicated roots.) Number Number of Positive Number of Negative Number of Positive Number of Negative Square Roots Square Roots Cube Roots Cube Roots State whether each of the following is true or false. a. A negative number has no real fourth roots. true b. represents both square roots of. true c. When you take the fifth root of, you must take the absolute value of to identify the principal fifth root. false Helping You Remember. What is an easy way to remember that a negative number has no real square roots but has one real cube root? Sample answer: The square of a positive or negative number is positive, so there is no real number whose square is negative. However, the cube of a negative number is negative, so a negative number has one real cube root, which is a negative number. NAME DATE PERIOD - Enrichment, p. 68 Approimating Square Roots Consider the following epansion. ( a b a ) a ab b a a a b b a b Think what happens if a is very great in comparison to b. The term a is very small and can be disregarded in an approimation. ( a b a ) a b b a a a b Suppose a number can be epressed as a b, a b. Then an approimate value b of the square root is a a. You should also see that a b a a. b Eample Use the formula b a b o appro e 0 an 6. 8 Chapter Polynomials

38 Space Science The escape velocity for the Moon is about 00 m/s. For the Sun, it is about 68,000 m/s. Source: NASA Standardized Test Practice Maintain Your Skills Mied Review 67. 7y (y y ) Getting Ready for the Net Lesson 6. SPACE SCIENCE The velocity v required for an object to escape the gravity of a planet or other body is given by the formula v G M, where M is the mass R of the body, R is the radius of the body, and G is Newton s gravitational constant. Use M.98 0 kg, R m, and G N m /kg to find the escape velocity for Earth. about,00 m/s 6. CRITICAL THINKING Under what conditions does y y? 0 and y 0, or y 0 and 0 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How do square roots apply to oceanography? Include the following in your answer: the values of s for,, and 0 feet, and an observation of what happens to the value of s as the value of increases. 6. Which of the following is closest to 7.? B.6 B.7 C.8 D In the figure, ABC is an equilateral triangle with sides 9 units long. What is the length of BD in units? D B 9 9 D 8 Factor completely. If the polynomial is not factorable, write prime. (Lesson -) 67. 7y y 8 y 68. ab a b (a )(b ) 69. ( )( ) 70. c 6 (c 6)(c 6c 6) Simplify. (Lesson -) ( 7 ) ( ) TRAVEL The matri at the right shows the costs of airline flights between some cities. Write a matri that shows the costs of two tickets for these flights. (Lesson -) Solve each system of equations by using either substitution or elimination. (Lesson -) 7. a b y 76. c 7d a b (, ) y (, ) c 6d 6 (9, ) Answer A A C PREREQUISITE SKILL Find each product. (To review multiplying binomials, see Lesson -.) New York LA Atlanta 0 60 Chicago ( )( 8) 78. (y )(y ) y y (a )(a 9) a 7a (a b)(a b) a ab b 8. ( y)( y) 9y 8. (w z)(w z) 6w 7wz z 6. The speed and length of a wave are related by an epression containing a square root. Answers should include the following. about.90 knots, about.00 knots, and. knots As the value of increases, the value of s increases. A B Lesson - Roots of Real Numbers 9 C 0 D About the Eercises Organization by Objective Simplify Radicals: 8 9 Approimate Radicals with a Calculator: 6 7 Odd/Even Assignments Eercises 6 9 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 7 odd, 9, 6, 6 8 Average: 7 6 odd, 6 8 Advanced: 6 6 even, 6 76 (optional: 77 8) Assess Open-Ended Assessment Modeling Have students use the hypotenuse of a right triangle with both legs one unit long to demonstrate and eplain that is a number on the real number line. Getting Ready for Lesson -6 PREREQUISITE SKILL Lesson -6 presents operations with radical epressions. In Eample on p., they will encounter the multiplication of two binomials involving radical epressions. Eercises 77 8 should be used to determine your students familiarity with multiplying binomials. Assessment Options Quiz (Lessons - and -) is available on p. 07 of the Chapter Resource Masters. Mid-Chapter Test (Lessons - through -) is available on p. 09 of the Chapter Resource Masters. Lesson - Roots of Real Numbers 9

39 Lesson Notes Radical Epressions Focus -Minute Check Transparency -6 Use as a quiz or a review of Lesson -. Mathematical Background notes are available for this lesson on p. 0D. Building on Prior Knowledge In Lesson -, students simplified radicals. In this lesson, students build on the skills they learned in that lesson to simplify and combine radical epressions. do radical epressions apply to falling objects? Ask students: If the value of d in the formula doubles, will the value of t also double? no Is the relationship between time t and distance d for a falling object linear or nonlinear? nonlinear Vocabulary rationalizing the denominator like radical epressions conjugates Simplify radical epressions. Add, subtract, multiply, and divide radical epressions. The amount of time t in seconds that it takes for an object to drop d feet is given by t d, where g ft/s g is the acceleration due to gravity. In this lesson, you will learn how to simplify radical epressions like d. g SIMPLIFY RADICAL EXPRESSIONS You can use the Commutative Property and the definition of square root to find an equivalent epression for a product of radicals such as. Begin by squaring the product. Commutative Property of Multiplication or Definition of square root Since 0 and, is the principal square root of. That is,. This illustrates the following property of radicals. Product Property of Radicals For any real numbers a and b and any integer n,. if n is even and a and b are both nonnegative, then n ab n a n b, and. if n is odd, then n ab n a n b. Follow these steps to simplify a square root. Step Factor the radicand into as many squares as possible. Step Use the Product Property to isolate the perfect squares. Step Simplify each radical. Eample do radical epressions apply to falling objects? Simplify 6p. 8 q 7 Square Root of a Product 6p 8 q 7 (p ) (q ) q (p ) (q ) q p q q Factor into squares where possible. Product Property of Radicals Simplify. However, for 6p 8 q 7 to be defined, 6p 8 q 7 must be nonnegative. If that is true, q must be nonnegative, since it is raised to an odd power. Thus, the absolute value is unnecessary, and 6p 8 q 7 p q q. Resource Manager 0 Chapter Polynomials Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp Skills Practice, p. 7 Practice, p. 7 Reading to Learn Mathematics, p. 7 Enrichment, p. 7 Teaching Algebra With Manipulatives Masters, p. 8 Transparencies -Minute Check Transparency -6 Real-World Transparency Answer Key Transparencies Technology Interactive Chalkboard

40 Study Tip Rationalizing the Denominator You may want to think of rationalizing the denominator as making the denominator a rational number. Look at a radical that involves division to see if there is a quotient property for radicals that is similar to the Product Property. Consider 9. The radicand is a 9 perfect square, so or 7. Notice that 7 9. This suggests the following property. 9 Quotient Property of Radicals Words For any real numbers a and b 0, and any integer n, n a n a, if all roots are defined. b n b Eample 7 9 or You can use the properties of radicals to write epressions in simplified form. Simplifying Radical Epressions A radical epression is in simplified form when the following conditions are met. The inde n is as small as possible. The radicand contains no factors (other than ) that are nth powers of an integer or polynomial. The radicand contains no fractions. No radicals appear in a denominator. To eliminate radicals from a denominator or fractions from a radicand, you can use a process called rationalizing the denominator. To rationalize a denominator, multiply the numerator and denominator by a quantity so that the radicand has an eact root. Study the eamples below. Eample Simplify Quotients Simplify each epression. a. y b. a y Quotient Property y a a ( ) Factor into squares. 8a (y y ) a 8a Quotient Property Rationalize the denominator. ( ) Product Property 8a Product Property (y ) y a 8a 0a ( ) Multiply. a y y y Rationalize the 0a a a y y y denominator. a Teach SIMPLIFY RADICAL EXPRESSIONS In-Class Eamples Power Point Teaching Tip When discussing the Product Property of Radicals, stress the fact that a and b must both be nonnegative if n is even. This means that times 8 may not be written as 6. This condition is necessary because and 8 are not real numbers. Simplify a. b 9 a b b Simplify each epression. y a. 8 7 b. 9 y 6 Teaching Tip Urge students to verify that each of their final answers is in simplified form by testing it against the four conditions listed in the Concept Summary for Simplifying Radical Epressions. y y y y y Lesson -6 Radical Epressions Differentiated Instruction Intrapersonal Have students think about irrational numbers, radicals, and the rules for operations with radicals. Ask them to write down what puzzles them most about these concepts, including a list of the definitions and operations about which they feel some confusion. Invite students to share these concerns with you so that they can be cleared up. Lesson -6 Radical Epressions

41 OPERATIONS WITH RADICALS In-Class Eample Power Point Simplify 00a 0a. 0a Teaching Tip After discussing the information presented in the Algebra Activity, make sure students also understand that a b is not equivalent to a. b Suggest they use the values a 6 and b 9 to verify this fact. OPERATIONS WITH RADICALS You can use the Product and Quotient Properties to multiply and divide some radicals, respectively. Eample Multiply Radicals Simplify 6 9n n. 6 9n n 6 9n n 8 n 8 n 8 n or 08n Product Property of Radicals Factor into cubes where possible. Product Property of Radicals Multiply. Can you add radicals in the same way that you multiply them? In other words, if a a a, a does a a a? a Adding Radicals You can use dot paper to show the sum of two like radicals, such as. Model and Analyze Step First, find a segment Step Etend the segment of length units by using to twice its length to represent the Pythagorean Theorem. with the dot paper. a b c c c. No; units is the length of the hypotenuse of an isosceles right triangle whose legs have length units. Therefore,. Study Tip Reading Math Indices is the plural of inde. Chapter Polynomials Make a Conjecture. Is or? Justify your answer using the geometric models above.. Use this method to model other irrational numbers. Do these models support your conjecture? See students work. In the activity, you discovered that you cannot add radicals in the same manner as you multiply them. You add radicals in the same manner as adding monomials. That is, you can add only the like terms or like radicals. Two radical epressions are called like radical epressions if both the indices and the radicands are alike. Some eamples of like and unlike radical epressions are given below. and are not like epressions. Different indices and are not like epressions. Different radicands a and a are like epressions. Radicands are a; indices are. Algebra Activity Materials: rectangular dot paper, ruler/straightedge Ask students what leg lengths they could use on a right triangle to find a line whose length is. lengths of and units Point out that there are other instances where you can perform a multiplication but not an addition. For eample, you can multiply fractions by multiplying the numerators and denominators separately, but you do not add fractions this way. Chapter Polynomials

42 Eample Add and Subtract Radicals Simplify Factor using squares. Product Property, 9 8 Multiply. Combine like radicals. In-Class Eamples Simplify Simplify each epression. a. ( )( ) Power Point b. ( 7)( 7) 7 Just as you can add and subtract radicals like monomials, you can multiply radicals using the FOIL method as you do when multiplying binomials. Eample Multiply Radicals a. F O I L 6 Product Property 6 6 or 6 b FOIL Multiply. 7 6 or 7 9 Subtract. 6 Simplify. 6 Binomials like those in Eample b, of the form ab cd and ab cd where a, b, c, and d are rational numbers, are called conjugates of each other. The product of conjugates is always a rational number. You can use conjugates to rationalize denominators. Eample 6 Use a Conjugate to Rationalize a Denominator Simplify. Multiply by because is the conjugate of. 8 6 FOIL Difference of squares Multiply. Combine like terms. Divide numerator and denominator by. Lesson -6 Radical Epressions Unlocking Misconceptions Radical Epressions When presented with a radical epression such as 6, some students may persist in trying to add the and the 6. Help them understand why this cannot be done by comparing this radical epression 6 to the epression 6. Stress that the radical 6 is a multiplication epression just like 6. Remind students that the order of operations requires that multiplication be performed before addition. Students may find it helpful to rewrite 6, as 6. Lesson -6 Radical Epressions

43 Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. add the information from the Key Concept and Concept Summary features to their notebook. include any other item(s) that they find helpful in mastering the skills in this lesson. Concept Check. Sometimes; n n a a only when a. Guided Practice GUIDED PRACTICE KEY Eercises Eamples, , 6 Application a b 0.. Determine whether the statement n a is sometimes, always, or never true. Eplain. n a. OPEN ENDED Write a sum of three radicals that contains two like terms.. Eplain why the product of two conjugates is always a rational number.. See margin. Simplify y y y y y ab 6a b LAW ENFORCEMENT A police accident investigator can use the formula s to estimate the speed s of a car in miles per hour based on the length in feet of the skid marks it left. How fast was a car traveling that left skid marks 0 feet long? about 9 mph About the Eercises Organization by Objective Simplify Radical Epressions: 0 Operations with Radicals: 8 Odd/Even Assignments Eercises 8 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: odd, 9, 8 Average: 9 odd, 0, 8 Advanced: 6 8 even, 0 7 (optional: 7 8) All: Practice Quiz ( 0) indicates increased difficulty Practice and Apply Homework Help For Eercises See Eamples Etra Practice See page 88.. c d c 6. wz wz Chapter Polynomials Simplify.. yy. ab 0a. 6y z 7. mn mn y y. 8 y. 0a b. 6y 6 z. m n. c d a 9. b 6. w6 z 7 r t What is 9 divided by 6?. Divide by. 0 Simplify a b b r t t Answers. Sample answer:. The product of two conjugates yields a difference of two squares. Each square produces a rational number and the difference of two rational numbers is a rational number. Chapter Polynomials 0. The square root of a difference is not the difference of the square roots. 6. The formula for the time it takes an object to fall a certain distance can be written in various forms involving radicals. Answers should include the following. By the Quotient Property of Radicals, t d g. Multiply by to rationalize the g g dg denominator. The result is. g about. s

44 Amusement Parks Attendance at the top 0 theme parks in North America increased to 7. million in 000. Source: Amusement Business 9. GEOMETRY Find the perimeter and area of the rectangle. 6 6 yd, 6 yd 6 yd AMUSEMENT PARKS For Eercises 0 and, use the following information. The velocity v in feet per second of a roller coaster at the bottom of a hill is related to the vertical drop h in feet and the velocity v 0 in feet per second of the coaster at the top of the hill by the formula v 0 vh Eplain why v 0 v 8h is not equivalent to the given formula. See margin.. What velocity must a coaster have at the top of a -foot hill to achieve a velocity of 0 feet per second at the bottom? 0 ft/s Online Research Data Update What are the values of v and h for some of the world s highest and fastest roller coasters? Visit to learn more. SPORTS For Eercises and, use the following information. A ball that is hit or thrown horizontally with a velocity of v meters per second will h travel a distance of d meters before hitting the ground, where d v.9 and h is the height in meters from which the ball is hit or thrown..9h. Use the properties of radicals to rewrite the formula. d v.9. How far will a ball that is hit horizontally with a velocity of meters per second at a height of 0.8 meter above the ground travel before hitting the ground? about 8.8 m. AUTOMOTIVE ENGINEERING An automotive engineer is trying to design a safer car. The maimum force a road can eert on the tires of the car being redesigned is 000 pounds. What is the maimum velocity v in ft/s at which Fc this car can safely round a turn of radius 0 feet? Use the formula v, r 0 0 where F c is the force the road eerts on the car and r is the radius of the turn. 80 ft/s or about mph. CRITICAL THINKING Under what conditions is the equation y y true? and y are nonnegative. 8 yd Study Study Guide and and Intervention Intervention, p. 69 (shown) and p NAME DATE PERIOD Radical Epressions Simplify Radical Epressions Product Property of Radicals For any real numbers a and b, and any integer n :. if n is even and a and b are both nonnegative, then ab n a n b. n. if n is odd, then ab n a n b. n To simplify a square root, follow these steps:. Factor the radicand into as many squares as possible.. Use the Product Property to isolate the perfect squares.. Simplify each radical. Quotient Property of Radicals n For any real numbers a and b 0, and any integer n, n a a nb, if all roots are defined. b To eliminate radicals from a denominator or fractions from a radicand, multiply the numerator and denominator by a quantity so that the radicand has an eact root. Eample Simplify 6a. b 7 Eample 6a b 7 () a a (b ) b Eercises ab a b Simplify.. 6. a 9 b 0 8 y 8 Simplify. y Quotient Property NAME 69 DATE PERIOD Practice (Average) a b a 8 y () (y ) y () (y ) y y y Factor into squares. Product Property Simplify. y Rationalize the y y y denominator. 0y Simplify. y. 7 y 7 y y.. 6. pq p 6 6 a 6 b a bb p q -6 Skills Practice, p. 7 and Practice, p. 7 (shown) Simplify t 6 w t w Radical Epressions v 8 z v z z g k 8 gk k 0. y 8 y c d 7 c d 9a 8 7a d a.. a 8 6b 9a 6 8b a 6. ()() 7. ()(78) ( ) Lesson WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How do radical epressions apply to falling objects? Include the following in your answer: an eplanation of how you can use the properties in this lesson to rewrite the. ( 7). ( 6)( ). ( 0)( 0) ( 6)( 7) 6. ( 7) 7. (08 6) Standardized Test Practice formula t, and gd the amount of time a -foot tall student has to get out of the way after a balloon is dropped from a window feet above. 7. The epression 80 is equivalent to which of the following? B A 6 B 6 C 0 D 6 8. Which of the following is not a length of a side of the triangle? D A 8 B C D NAME DATE PERIOD -6 Enrichment, p. 7 Special Products with Radicals Notice that ()(), or (). In general, () when 0. Also, notice that (9)() 6. In general, ()(y) y when and y are not negative. You can use these ideas to find the special products below. (a b)(a b) (a) (b) a b (a b) (a) ab (b) a ab b (a b) (a) ab (b) a ab b Eample Find the product: ( )( ). ( )( ) () () Eample Lesson -6 Radical Epressions Evaluate ( 8). 6. BRAKING The formula s estimates the speed s in miles per hour of a car when it leaves skid marks feet long. Use the formula to write a simplified epression for s if 8. Then evaluate s to the nearest mile per hour. 07 ; mi/h. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can be represented by the epressions 6 y and 9 y. Use the Pythagorean Theorem to find a simplified epression for the measure of the hypotenuse. y Reading to Learn Mathematics, p. 7-6 NAME 7 DATE PERIOD Reading to Learn Mathematics Radical Epressions ELL Pre-Activity How do radical epressions apply to falling objects? Read the introduction to Lesson -6 at the top of page 0 in your tetbook. Describe how you could use the formula given in your tetbook and a calculator to find the time, to the nearest tenth of a second, that it would take for the water balloons to drop feet. (Do not actually calculate the time.) Sample answer: Multiply by (giving ) and divide by. Use the calculator to find the square root of the result. Round this square root to the nearest tenth. Reading the Lesson. Complete the conditions that must be met for a radical epression to be in simplified form. The inde n is as small as possible. The radicand contains no factors (other than ) that are nth powers of a(n) integer or polynomial. The radicand contains no fractions. No radicals appear in the denominator.. a. What are conjugates of radical epressions used for? to rationalize binomial denominators b. How would you use a conjugate to simplify the radical epression? Multiply numerator and denominator by. c. In order to simplify the radical epression in part b, two multiplications are necessary. The multiplication in the numerator would be done by the FOIL method, and the multiplication in the denominator would be done by finding the difference of two squares. Helping You Remember. One way to remember something is to eplain it to another person. When rationalizing the denominator in the epression, many students think they should multiply numerator and denominator by. How would you eplain to a classmate why this is incorrect and what he should do instead. Sample answer: Because you are working with cube roots, not square roots, you need to make the radicand in the denominator a perfect cube, not a perfect square. Multiply numerator and denominator by to make the denominator 8, which equals. Lesson -6 Radical Epressions Lesson -6

45 Assess Open-Ended Assessment Speaking Ask students to describe how combining radicals is the same as combining epressions with variables, and how it differs from working with variables. Maintain Your Skills Mied Review Simplify. (Lesson -) 9. z 8 z 60. 6a b 9 6ab 6. (y ) y Simplify. Assume that no denominator is equal to 0. (Lesson -) Perform the indicated operations. (Lesson -) Intervention Students will need to simplify epressions involving radicals in much of their further work in algebra. Take time to help students uncover and correct their misconceptions by analyzing the errors they make. New Getting Ready for Lesson -7 BASIC SKILL Lesson -7 presents working with rational eponents. This often involves adding, subtracting, or multiplying fractions. Eercises 7 8 should be used to determine your students familiarity with rational numbers. Getting Ready for the Net Lesson 66. Find the maimum and minimum values of the function f(, y) y for the region with vertices at (, ), (, ), (, ), and (, ). (Lesson -) 6, 67. State whether the system of equations shown at the right is consistent and independent, consistent and dependent, or inconsistent. (Lesson -) consistent and independent 68. BUSINESS The amount that a mail-order company charges for shipping and handling is given by the function c() 0., where is the weight in pounds. Find the charge for an 8-pound order. (Lesson -) $.0 Solve. (Lessons -, -, and -) , 7. 7, 7. 8 { 6} 7. { 7} BASIC SKILL Evaluate each epression y O Assessment Options Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons - through -6. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not yet mastered. P ractice Quiz Factor completely. If the polynomial is not factorable, write prime. (Lesson -). y y y y( y ). prime. a 6a 9a a( ). 8r 6s 6 8(r s )(r rs s ) Simplify. (Lessons - and -6). 6 y 6 6 y 6. 6a b 6 9 a b 8. y y y Lessons - through n n 9 n 8 6 Chapter Polynomials 6 Chapter Polynomials

46 Rational Eponents Lesson Notes Write epressions with rational eponents in radical form, and vice versa. Simplify epressions in eponential or radical form. do rational eponents apply to astronomy? Astronomers refer to the space around a planet where the planet s gravity is stronger than the Sun s as the sphere of influence of the planet. The radius r of the sphere of influence is given by Mp, where M p is the mass the formula r D M S of the planet, M S is the mass of the Sun, and D is the distance between the planet and the Sun. RATIONAL EXPONENTS AND RADICALS You know that squaring a number and taking the square root of a number are inverse operations. But how would you evaluate an epression that contains a fractional eponent such as the one above? You can investigate such an epression by assuming that fractional eponents behave as integral eponents. b b b Write the square as multiplication. b Add the eponents. b or b Simplify. Thus, b is a number whose square equals b. So it makes sense to define b b. Words For any real number b and for any positive integer n, b b n n n b, ecept when b 0 and n is even. Eample 8 8 or D r Focus -Minute Check Transparency -7 Use as a quiz or review of Lesson -6. Mathematical Background notes are available for this lesson on p. 0D. do rational eponents apply to astronomy? Ask students: Is the p in M p an eponent? Is it a variable? No, it is neither an eponent nor a variable; it is a subscript. Would you epect the radius of the sphere of influence for one of the larger planets in our solar system to be greater than the radius of the sphere of influence for Earth? Use the formula to justify your answer. Yes; for the planets larger than Earth, the value of M p would be greater than the value of M p for Earth while the value of M S is the same. So the value of M p the ratio is greater for the MS larger planets. Eample Radical Form Write each epression in radical form. a. a b. a a Definition of b n Definition of b n Lesson -7 Rational Eponents 7 Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp Skills Practice, p. 77 Practice, p. 78 Reading to Learn Mathematics, p. 79 Enrichment, p. 80 Assessment, p. 08 Graphing Calculator and Spreadsheet Masters, p. 6 Transparencies -Minute Check Transparency -7 Answer Key Transparencies Technology Interactive Chalkboard Lesson - Lesson Title 7

47 Teach RATIONAL EXPONENTS AND RADICALS In-Class Eamples Write each epression in radical form. a. a 6 6 a b. m m Write each radical using rational eponents. a. b b b. w w Power Point Evaluate each epression. a. 9 7 b. Teaching Tip If students are having difficulty remembering which part of the fractional eponent is the inde, suggest that they recall the basic definition b b. Study Tip Negative Base Suppose the base of a monomial is negative such as (9) or (9). The epression is undefined if the eponent is even because there is no number that, when multiplied an even number of times, results in a negative number. However, the epression is defined for an odd eponent. Eample Eample Evaluate Epressions with Rational Eponents Evaluate each epression. a. 6 Method Method 6 b n b 6 ( ) 6 b. 6 Eponential Form Write each radical using rational eponents. a. y y y Definition of b n b. 8 c 8 c c 8 Definition of b n Many epressions with fractional eponents can be evaluated using the definition of b n or the properties of powers. n Power of a Power 6 Multiply eponents. Simplify. Method Method Factor. ( ) ( ) Power of a Power Power of a Power b b Multiply eponents. ( ) 7 Epand the cube. or 7 Find the fifth root. In Eample b, Method uses a combination of the definition of b n and the properties of powers. This eample suggests the following general definition of rational eponents. Rational Eponents Words For any nonzero real number b, and any integers m and n, with n, b m n n b m n b m, ecept when b 0 and n is even. Eample or 8 Chapter Polynomials In general, we define b m n b n to bn m and (b m ) n. bn m n b m (b m ) n n b m as bn m or (b m ) n. Now apply the definition of Unlocking Misconceptions Eponents Students may be confused because they are not perceiving and reading the eponent in a way that distinguishes it from a coefficient or multiplier. Ask them to practice reading the eponent correctly, for eample, reading as to the third power or as cubed.. Radicals Ask students to practice reading radical epressions correctly, for eample, reading y as the square root of y cubed. 8 Chapter Polynomials

48 Weight Lifting With origins in both the ancient Egyptian and Greek societies, weightlifting was among the sports on the program of the first Modern Olympic Games, in 896, in Athens, Greece. Source: International Weightlifting Association TEACHING TIP Tell students that if they are simplifying an epression that was originally written with radicals, they should write the answer with radicals. If the epression was originally written with rational eponents, they should write the answer with rational eponents. Eample SIMPLIFY EXPRESSIONS All of the properties of powers you learned in Lesson - apply to rational eponents. When simplifying epressions containing rational eponents, leave the eponent in rational form rather than writing the epression as a radical. To simplify such an epression, you must write the epression with all positive eponents. Furthermore, any eponents in the denominator of a fraction must be positive integers. So, it may be necessary to rationalize a denominator. Rational Eponent with Numerator Other Than WEIGHT LIFTING The formula M 6,0B 8 can be used to estimate the maimum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and the clean and jerk, combined. a. According to the formula, what is the maimum amount that 000 Olympic champion Xugang Zhan of China can lift if he weighs 7 kilograms? M 6,0B 8 Original formula 6,0(7) 8 B 7 6 kg Use a calculator. The formula predicts that he can lift at most 6 kilograms. b. Xugang Zhan s winning total in the 000 Olympics was 67.0 kg. Compare this to the value predicted by the formula. The formula prediction is close to the actual weight, but slightly lower. Eample Simplify each epression. a. 7 Simplify Epressions with Rational Eponents 7 7 Multiply powers. b. y 8 Add eponents. y b n b n y y Why use? y y y y y y y y y y y y or y y When simplifying a radical epression, always use the smallest inde possible. Using rational eponents makes this process easier, but the answer should be written in radical form. Lesson -7 Rational Eponents 9 In-Class Eample In-Class Eample Power Point Teaching Tip Be sure students notice that the fractional eponent is negative in the formula for the maimum total mass M. WEIGHT LIFTING Use the formula given in Eample. a. U.S. weightlifter Oscar Chaplin III competed in the same weight class as Xugang Zhan, finishing in 7th place. According to the formula, what is the maimum that Chaplin can lift if he weighs 77 kilograms? Source: cnnsi.com The formula predicts that he can lift at most 7 kilograms. b. Oscar Chaplin s total in the 000 Olympics was kg. Compare this to the value predicted by the formula. The formula prediction is somewhat higher than his actual total. SIMPLIFY EXPRESSIONS Simplify each epression. a. y 7 y 7 y 7 b. Power Point Differentiated Instruction Auditory/Musical Have students name and demonstrate on a keyboard, guitar, or other instrument, the sounds of the notes described in Eercises 67 and 68. Other students can work to associate the sound of the note with the number of vibrations per second given by the formula. Lesson -7 Rational Eponents 9

49 In-Class Eample Practice/Apply Study Notebook Power Point 6 Simplify each epression. 6 6 a. or b. c. 6 y y y y y Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. add the information listed in the Concept Summary below Eample 6 to their notebook. include any other item(s) that they find helpful in mastering the skills in this lesson. About the Eercises Organization by Objective Rational Eponents and Radicals: 0 Simplify Epressions: 6 Odd/Even Assignments Eercises 66 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 6 odd, 6, 69 8 Average: 6 odd, 69 8 Advanced: 66 even, 67, 68, (optional: 8 8) Concept Check. Sample answer: 6 60 Chapter Polynomials Answers Eample 6 Simplify Radical Epressions Simplify each epression. a Rational eponents b. 9z. In radical form, the epression would be 6, which is not a real number because the inde is even and the radicand is negative. c. Power of a Power 6 Quotient of Powers or Simplify. 9z (9z ) m m m m 6 ( z ) 9 z z Multiply. Rational eponents Power of a Power z, z z z Simplify. m m m is the conjugate of m. m m m m m Multiply.. OPEN ENDED Determine a value of b for which b 6 is an integer.. Eplain why (6) is not a real number. See margin.. Eplain why n b m n b m. See margin. Epressions with Rational Eponents An epression with rational eponents is simplified when all of the following conditions are met. It has no negative eponents. It has no fractional eponents in the denominator. It is not a comple fraction. The inde of any remaining radical is the least number possible.. In eponential form b n m is equal to (b m ) n. By the Power of a Power Property, (b m ) n b m n. But, b m n is also equal to (b n) m by the Power of a Power Property. This last epression is equal to ( n b) m. Thus, n b m ( n b) m. 60 Chapter Polynomials

50 Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6, , z( y) 7. y Application Practice and Apply Homework Help For See Eercises Eamples 8 9 0, Etra Practice See page 88. Write each epression in radical form or Write each radical using rational eponents y 7 6 y 7 Evaluate each epression Simplify each epression.. a a a. z 6 a b.. a b z Write each epression in radical form c. ( ) c or c Write each radical using rational eponents z z 8. y y. a a 9. w 6. 6 w 7. t 8. r t r 9. t y y y c 6 c c y c y c (mn ) m n 7. z( y) mn y y a b 8 b 9a b 9. y y z ab 60. z z c ab c c z z z 6 r 6 w NAME DATE PERIOD -7 Enrichment, p. 80 Lesser-Known Geometric Formulas Many geometric formulas involve radical epressions. Make a drawing to illustrate each of the formulas given on this page. Then evaluate the formula for the given value of the variable. Round answers to the nearest hundredth.. The area of an isosceles triangle. Two sides have length a; the other side has length c. Find A when a 6 and c 7. c A a c a a A 7.06 c. The area of a regular pentagon with a side of length a. Find A when a. a 0 a a ECONOMICS When inflation causes the price of an item to increase, the new cost C and the original cost c are related by the formula C c( r) n, where r is the rate of inflation per year as a decimal and n is the number of years. What would be the price of a $.99 item after si months of % inflation? $. indicates increased difficulty Evaluate each epression (7) 9. () Simplify each epression.. y y 7 y. 9. b b z a 6 6a a 9 b 7 a 6a a Lesson -7 Rational Eponents 6. The area of an equilateral triangle with a side of length a. Find A when a 8. A a A 7.7 a a. The area of a regular heagon with a side of length a. Find A when a 9. a A a a Study Study Guide and and Intervention Intervention, p. 7 (shown) and p NAME DATE PERIOD Rational Eponents Rational Eponents and Radicals Definition of b For any real number b and any positive integer n, n b n b, n ecept when b 0 and n is even. Definition of b m For any nonzero real number b, and any integers m and n, with n, n b m n b n m ( b) n m, ecept when b 0 and n is even. Write 8 Eample in radical form. Eample 8 Evaluate. Notice that 8 0. Notice that 8 0, 0, and is odd Eercises Write each epression in radical form Write each radical using rational eponents.. 7. a b 6. 6p 7 a b p Evaluate each epression (0.000) (0.) -7 NAME 7 DATE PERIOD Practice (Average) Skills Practice, p. 77 and Practice, p. 78 (shown) Write each epression in radical form m 7. (n ) 6 or ( 6) m 7 or ( m) 7 Write each radical using rational eponents m 6 n n n 8. a 0 b 79 m n a b Evaluate each epression (6) Rational Eponents Simplify each epression. 8. g 7 g 7 g 9. s s s 0. u u 6 9. y y y. b. q z z.. z t q b t b z q t t z ab b 0. ELECTRICITY The amount of current in amperes I that an appliance uses can be calculated using the formula I P, where P is the power in watts and R is the R resistance in ohms. How much current does an appliance use if P 00 watts and R 0 ohms? Round your answer to the nearest tenth. 7. amps. BUSINESS A company that produces DVDs uses the formula C 88n 0 to calculate the cost C in dollars of producing n DVDs per day. What is the company s cost to produce 0 DVDs per day? Round your answer to the nearest dollar. $798 Reading to Learn Mathematics, p a b NAME 78 DATE PERIOD Reading to Learn Mathematics Rational Eponents Pre-Activity How do rational eponents apply to astronomy? Read the introduction to Lesson -7 at the top of page 7 in your tetbook. The formula in the introduction contains the eponent. What do you think it might mean to raise a number to the power? Sample answer: Take the fifth root of the number and square it. Reading the Lesson. Complete the following definitions of rational eponents. For any real number b and for any positive integer n, b n n b ecept when b 0 and n is even. For any nonzero real number b, and any integers m and n, with n, b m n b n m ( b) n m, ecept when b 0 and n is even.. Complete the conditions that must be met in order for an epression with rational eponents to be simplified. It has no negative eponents. It has no fractional eponents in the denominator. It is not a comple fraction. The inde of any remaining radical is the least number possible.. Margarita and Pierre were working together on their algebra homework. One eercise asked them to evaluate the epression 7. Margarita thought that they should raise 7 to the fourth power first and then take the cube root of the result. Pierre thought that they should take the cube root of 7 first and then raise the result to the fourth power. Whose method is correct? Both methods are correct. Helping You Remember. Some students have trouble remembering which part of the fraction in a rational eponent gives the power and which part gives the root. How can your knowledge of integer eponents help you to keep this straight? Sample answer: An integer eponent can be written as a rational eponent. For eample,. You know that this means that is raised to the third power, so the numerator must give the power, and, therefore, the denominator must give the root. ELL Lesson -7 Rational Eponents 6 Lesson -7 Lesson -7

51 Assess Open-Ended Assessment Speaking Have students write two epressions with rational eponents, one that is in simplified form and another that is not. Ask them to eplain the difference between them, using the four conditions listed in the Concept Summary on p. 60. Intervention In order to help students see why the eception ecept when b 0 and n is even is necessary when defining rational eponents, ask them to choose values for b and n that violate these constraints, and see what results when applying the definition. New Getting Ready for Lesson -8 PREREQUISITE SKILL Lesson -8 presents solving equations and inequalities that contain radicals. Solving such equations and inequalities involves finding the power of an epression involving a radical. Eercises 8 8 should be used to determine your students familiarity with multiplying radicals. Assessment Options Quiz (Lessons -6 and -7) is available on p. 08 of the Chapter Resource Masters. Music The first piano was made in about 709 by Bartolomeo Cristofori, a maker of harpsichords in Florence, Italy. Source: Standardized Test Practice Maintain Your Skills Mied Review Getting Ready for the Net Lesson 6 Chapter Polynomials 6. Find the simplified form of What is the simplified form of 8? MUSIC For Eercises 67 and 68, use the following information. On a piano, the frequency of the A note above middle C should be set at 0 vibrations per second. The frequency f n of a note that is n notes above that A should be f n 0 n. 67. At what frequency should a piano tuner set the A that is one octave, or notes, above the A above middle C? 880 vibrations per second 68. Middle C is nine notes below the A that has a frequency of 0 vibrations per second. What is the frequency of middle C? about 6 vibrations per second 69. BIOLOGY Suppose a culture has 00 bacteria to begin with and the number of bacteria doubles every hours. Then the number N of bacteria after t hours is given by N 00 t. How many bacteria will be present after and a half hours? about CRITICAL THINKING Eplain how to solve 9 for. See margin. 7. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 8A 8B. How do rational eponents apply to astronomy? Include the following in your answer: an eplanation of how to write the formula r D M simplify it, and Mp S in radical form and an eplanation of what happens to the value of r as the value of D increases assuming that M p and M S are constant. 7. Which is the value of? C A B C 6 7. If y and y, then what is the value of y? C A B C D 6 Simplify. (Lessons - and -6) 7. y y (8) ( ) BIOLOGY Humans blink their eyes about once every seconds. How many times do humans blink their eyes in two hours? (Lesson -) 0 PREREQUISITE SKILL Find each power. (To review multiplying radicals, see Lesson -6.) D Answer 70. Rewrite the equation so that the bases are the same on each side. 9 ( ) Since the bases are the same and this is an equation, the eponents must be equal. Solve. The result is. 6 Chapter Polynomials

52 Radical Equations and Inequalities Lesson Notes Vocabulary radical equation etraneous solution radical inequality Solve equations containing radicals. Solve inequalities containing radicals. do radical equations apply to manufacturing? Computer chips are made from the element silicon, which is found in sand. Suppose a company that manufactures computer chips uses the formula C 0n 00 to estimate the cost C in dollars of producing n chips. This equation can be rewritten as a radical equation. SOLVE RADICAL EQUATIONS Equations with radicals that have variables in the radicands are called radical equations. To solve this type of equation, raise each side of the equation to a power equal to the inde of the radical to eliminate the radical. Eample Solve a Radical Equation Solve. Original equation Subtract from each side to isolate the radical. Square each side to eliminate the radical. Find the squares. Subtract from each side. CHECK Original equation Replace with. Simplify. The solution checks. The solution is. When you solve a radical equation, it is very important that you check your solution. Sometimes you will obtain a number that does not satisfy the original equation. Such a number is called an etraneous solution. You can use a graphing calculator to predict the number of solutions of an equation or to determine whether the solution you obtain is reasonable. Eample Etraneous Solution Solve Original equation Square each side. Find the squares. Isolate the radical. Divide each side by 6. Square each side again. 6 Evaluate the squares. (continued on the net page) Lesson -8 Radical Equations and Inequalities 6 Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp. 8 8 Skills Practice, p. 8 Practice, p. 8 Reading to Learn Mathematics, p. 8 Enrichment, p. 86 Focus -Minute Check Transparency -8 Use as a quiz or review of Lesson -7. Mathematical Background notes are available for this lesson on p. 0D. do radical equations apply to manufacturing? Ask students: Why can the equation be rewritten as a radical equation? because the variable n has a rational eponent Manufacturing There are production costs associated with manufactured goods that occur even before the first item is made. That is, there is still a cost even if no items have been produced yet. How much are these costs for the production of this company s computer chips? $00 Resource Manager Transparencies -Minute Check Transparency -8 Answer Key Transparencies Technology AlgePASS: Tutorial Plus, Lesson 9 Interactive Chalkboard Lesson - Lesson Title 6

53 Teach SOLVE RADICAL EQUATIONS In-Class Eamples Power Point Solve y. 8 Solve. no solution Teaching Tip Remind students that the square root sign in an equation means the principal root. Solve (y ) 0. Teaching Tip Have a discussion with students about which operations may introduce etraneous solutions when solving a radical equation. Study Tip Alternative Method To solve a radical equation, you can substitute a variable for the radical epression. In Eample, let A n. A 0 A A 8 A 7 8 n 7 7 n 7 CHECK 6 6 The solution does not check, so the equation has no real solution. The graphing calculator screen shows the graphs of y and y. The graphs do not intersect, which confirms that there is no solution. [0, 0] scl: by [, ] scl: You can apply the same methods used in solving square root equations to solving equations with roots of any inde. Remember that to undo a square root, you square the epression. To undo an nth root, you must raise the epression to the nth power. Eample Solve (n ) 0. Cube Root Equation In order to remove the power, or cube root, you must first isolate it and then raise each side of the equation to the third power. (n ) 0 Original equation (n ) Add to each side. (n ) Divide each side by. (n ) Cube each side. 8 n 7 Evaluate the cubes. n 7 Add to each side. 7 n 7 Divide each side by. CHECK (n ) 0 Original equation Replace n with Simplify. 7 7 The solution is The cube root of 7 is. 0 0 Subtract. 6 Chapter Polynomials SOLVE RADICAL INEQUALITIES You can use what you know about radical equations to help solve radical inequalities. A radical inequality is an inequality that has a variable in a radicand. 6 Chapter Polynomials

54 Study Tip Check Your Solution You may also want to use a graphing calculator to check. Graph each side of the original inequality and eamine the intersection. Concept Check Eample To solve radical inequalities, complete the following steps. Step Step Step Radical Inequality Solve 6. Since the radicand of a square root must be greater than or equal to zero, first solve 0 to identify the values of for which the left side of the given inequality is defined. 0 Now solve 6. 6 Original inequality Isolate the radical. 6 Eliminate the radical. 0 Add to each side. Divide each side by. It appears that. You can test some values to confirm the solution. Let f(). Use three test values: one less than, one between and, and one greater than. Organize the test values in a table. 0 7 f(0) (0) f() () f(7) (7) 6.90 Since is not a Since 6, the Since , the real number, the inequality is satisfied. inequality is not inequality is not satisfied. satisfied. The solution checks. Only values in the interval satisfy the inequality. You can summarize the solution with a number line Solving Radical Inequalities If the inde of the root is even, identify the values of the variable for which the radicand is nonnegative. Solve the inequality algebraically. Test values to check your solution.. Eplain why you do not have to square each side to solve. Then solve the equation. See margin.. Show how to solve by factoring. Name the properties of equality that you use. See margin.. OPEN ENDED Write an equation containing two radicals for which is a solution. Sample answer: Differentiated Instruction Lesson -8 Radical Equations and Inequalities 6 Logical Have students compare solving radical equations and inequalities to solving other types of equations and inequalities. Have them write or give a short presentation about the similarities and differences between the procedures used in the solution processes. SOLVE RADICAL INEQUALITIES In-Class Eample Solve 6 7. Practice/Apply Study Notebook Power Point Teaching Tip Emphasize the importance of checking key test values in the appropriate ranges. Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter. write a list of the steps for solving radical equations and copy the list of steps for solving radical inequalities given in the Concept Summary on p. 6. include any other item(s) that they find helpful in mastering the skills in this lesson. Answers. Since is not under the radical, the equation is a linear equation, not a radical equation. The solution is.. The trinomial is a perfect square in terms of. 6 9 ( ), so the equation can be written as ( ) 0. Take the square root of each side to get 0. Use the Addition Property of Equality to add to each side, then square each side to get 9. Lesson -8 Radical Equations and Inequalities 6

55 Study Study Guide and and Intervention Intervention, p. 8 (shown) and p. 8-8 NAME DATE PERIOD Radical Equations and Inequalities Solve Radical Equations The following steps are used in solving equations that have variables in the radicand. Some algebraic procedures may be needed before you use these steps. Step Step Step Step Isolate the radical on one side of the equation. To eliminate the radical, raise each side of the equation to a power equal to the inde of the radical. Solve the resulting equation. Check your solution in the original equation to make sure that you have not obtained any etraneous roots. Eample Solve 8 8. Eample Solve. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 9, 0, Solve each equation or inequality... (7 y) no solution 7. z (a) b b 0 b Original equation Add to each side. Isolate the radical. Square each side. Subtract 8 from each side. 7 Divide each side by. Check (7) (6) The solution 7 checks. Eercises Original equation Square each side. 0 ( ) 0 Simplify. Isolate the radical. Square each side. Subtract from each side. Factor. 0 or Check (0), but (0), so 0 is not a solution. (), and (), so the solution is. Lesson -8 Application. GEOMETRY The surface area S of a cone can be found by using S rr h, where r is the radius of the base and h is the height of the cone. Find the height of the cone. about. cm h S cm r cm Solve each equation no solution no solution no solution , -8 NAME 8 DATE PERIOD Practice (Average) Skills Practice, p. 8 and Practice, p. 8 (shown) Radical Equations and Inequalities Solve each equation or inequality p 0. h 0. c h no solution 7. d 7 8. w q 9 0. y 9 0 no solution. m n. t m 8 6. t 6. (7v ) 7 no solution 7. (g ) 6 8. (6u ) 0 9. d d 0. r 6 r no solution. a a 6. z z q no solution 6. a a 7. 9 c 6 c STATISTICS Statisticians use the formula v to calculate a standard deviation, where v is the variance of a data set. Find the variance when the standard deviation is. 0. GRAVITATION Helena drops a ball from feet above a lake. The formula t h describes the time t in seconds that the ball is h feet above the water. How many feet above the water will the ball be after second? 9 ft Reading to Learn Mathematics, p. 8-8 NAME 8 DATE PERIOD Reading to Learn Mathematics Radical Equations and Inequalities Pre-Activity How do radical equations apply to manufacturing? Read the introduction to Lesson -8 at the top of page 6 in your tetbook. Eplain how you would use the formula in your tetbook to find the cost of producing,000 computer chips. (Describe the steps of the calculation in the order in which you would perform them, but do not actually do the calculation.) Sample answer: Raise,000 to the power by taking the cube root of,000 and squaring the result (or raise,000 to the power by entering,000 ^ (/) on a calculator). Multiply the number you get by 0 and then add 00. Reading the Lesson. a. What is an etraneous solution of a radical equation? Sample answer: a number that satisfies an equation obtained by raising both sides of the original equation to a higher power but does not satisfy the original equation 6 ELL indicates increased difficulty Practice and Apply Homework Help For Eercises See Eamples, 9, 7 8, 6 Etra Practice See page Chapter Polynomials Solve each equation or inequality.. 6. y a 9 0 no solution 6. z 0 no solution 7. c 9 8. m y 7. (6n ) 0. ( 7). no solution. t 7 t y y 9. 6 no solution 0. y y. b b 6. z z no solution a 9 a 0 a. b b 7 b 6. c c 0. c What is the solution of 6? 8. Solve CONSTRUCTION The minimum depth d in inches of a beam required s to support a load of s pounds is given by the formula d, 7 6w where is the length of the beam in feet and w is the width in feet. Find the load that can be supported by a board that is feet long, feet wide, and inches deep. lb 0. AEROSPACE ENGINEERING The radius r of the orbit of a satellite is given by r G Mt, where G is the universal gravitational constant, M is the mass of the central object, and t is the time it takes the satellite to complete one orbit. Solve this formula for t. t r GM b. Describe two ways you can check the proposed solutions of a radical equation in order to determine whether any of them are etraneous solutions. Sample answer: One way is to check each proposed solution by substituting it into the original equation. Another way is to use a graphing calculator to graph both sides of the original equation. See where the graphs intersect. This can help you identify solutions that may be etraneous.. Complete the steps that should be followed in order to solve a radical inequality. Step If the inde of the root is even, identify the values of the variable for which the radicand is nonnegative. Step Solve the inequality algebraically. Step Test values to check your solution. Helping You Remember. One way to remember something is to eplain it to another person. Suppose that your friend Leora thinks that she does not need to check her solutions to radical equations by substitution because she knows she is very careful and seldom makes mistakes in her work. How can you eplain to her that she should nevertheless check every proposed solution in the original equation? Sample answer: Squaring both sides of an equation can produce an equation that is not equivalent to the original one. For eample, the only solution of is, but the squared equation has two solutions, and. NAME DATE PERIOD -8 Enrichment, p. 86 Truth Tables In mathematics, the basic operations are addition, subtraction, multiplication, division, finding a root, and raising to a power. In logic, the basic operations are the following: not (), and (), or (), and implies ( ). If P and Q are statements, then P means not P; Q means not Q; P Q means P and Q; P Q means P or Q; and P Q means P implies Q. The operations are defined by truth tables. On the left below is the truth table for the statement P. Notice that there are two possible conditions for P, true (T) or false (F). If P is true, P is false; if P is false, P is true. Also shown are the truth tables for P Q, P Q, and P Q. P P P Q P Q P Q P Q P Q P Q T F T T T T T T T T T F T T F F T F T T F F F T F F T T F T T F F F F F F F F T You can use this information to find out under what conditions a comple ment 66 Chapter Polynomials

56 Health A ponderal inde p is a measure of a person s body based on height h in meters and mass m in kilograms. One such formula is p m. h Source: A Dictionary of Food and Nutrition Standardized Test Practice. PHYSICS When an object is dropped from the top of a 0-foot tall building, the object will be h feet above the ground after t seconds, where 0 h t. How far above the ground will the object be after second? ft. HEALTH Use the information about health at the left. A 70-kilogram person who is.8 meters tall has a ponderal inde of about.9. How much weight could such a person gain and still have an inde of at most.?. kg. CRITICAL THINKING Eplain how you know that has no real solution without trying to solve it. See margin.. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 8A 8B. How do radical equations apply to manufacturing? Include the following in your answer: the equation C 0n 00 rewritten as a radical equation, and a step-by-step eplanation of how to determine the maimum number of chips the company could make for $0,000.. If, what is the value of? D A B 0 C D About the Eercises Organization by Objective Solve Radical Equations:, 9, 7 Solve Radical Inequalities: 8, 6 Odd/Even Assignments Eercises 8 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 7 odd, 7 odd, 9 Average: odd, 9 Advanced: even, (optional: 9) Maintain Your Skills Mied Review 6. Side AC of triangle ABC contains which of the following points? C A (, ) B (, ) C (, ) D (, ) E (, 6) y A(0, 0) O Write each radical using rational eponents. (Lesson -7) ( 7) 9. ( ) B 8 0 C Assess Open-Ended Assessment Writing Have students write a list of eamples showing how to solve each of the different types of radical equations and inequalities discussed in this lesson. Getting Ready for the Net Lesson Simplify. (Lesson -6) y 6 yy. Answers BUSINESS A dry cleaner ordered 7 drums of two different types of cleaning fluid. One type cost $0 per drum, and the other type cost $0 per drum. The total cost was $60. How much of each type of fluid did the company order? Write a system of equations and solve by graphing. (Lesson -) y 7, 0 0y 60; See margin for graph; (, ). PREREQUISITE SKILL Simplify each epression. (To review binomials, see Lesson -.). ( ) ( ). ( y) ( y) y 6. ( ) ( ) 7. (7 ) ( ) 8. ( z)( z) 6z z 9. ( )( ) 0 8. Since 0 and 0, the left side of the equation is nonnegative. Therefore, the left side of the equation cannot equal. Thus, the equation has no solution.. y Lesson -8 Radical Equations and Inequalities y 60 (, ) y 7 Intervention Make sure that students know the constraints on the values of the variables in a radical equation so that the solutions are real numbers. New Getting Ready for Lesson -9 PREREQUISITE SKILL Lesson -9 presents calculating with comple numbers, often written in the form of binomials. Eercises 9 should be used to determine your students familiarity with adding, subtracting, and multiplying binomials. O Lesson -8 Radical Equations and Inequalities 67

57 Graphing Calculator Investigation A Follow-Up of Lesson -8 Getting Started Know Your Calculator The TI-8 Plus automatically supplies a left parenthesis after each radical sign. When functions are entered on the Y= list, it is important to supply right parentheses as needed to ensure correct graphs and correct numerical results. Displaying Tables In Step on p. 68, students should check to be sure that the AUTO option has been selected on each of the last two lines of the TABLE SETUP screen. Eact Solutions The approimate zero displayed on the screen shown in Step on p. 68 appears to be a repeating decimal. If you go to the home screen immediately after Step and use the keystrokes X,T,,n ENTER, the calculator will display a 9 fraction for the eact solution,. 6 The calculator can be used to verify that this is indeed the eact solution of the equation. Teach After reading the sentence at the top of p. 69, have students solve the radical equation on p. 68 again treating each side as a separate function. Point out that the right side will simply be graphed as the function y. After completing the discussion of the procedure on p. 69 for solving a radical inequality, have students solve it again by first subtracting from both sides and then graphing the function y. Point out that the portion of the graph below the -ais shows the solution. Have students complete Eercises 0. Solving Radical Equations and Inequalities by Graphing You can use a TI-8 Plus to solve radical equations and inequalities. One way to do this is by rewriting the equation or inequality so that one side is 0 and then using the zero feature on the calculator. Solve. 68 Chapter Polynomials Rewrite the equation. Subtract from each side of the equation to obtain 0. Enter the function y in the Y= list. KEYSTROKES: Review entering a function on page 8. Estimate the solution. Complete the table and estimate the solution(s). KEYSTROKES: nd [TABLE] Since the function changes sign from negative to positive between and, there is a solution between and. Use a table. A Follow-Up of Lesson -8 You can use the TABLE function to locate intervals where the solution(s) lie. First, enter the starting value and the interval for the table. KEYSTROKES: nd [TBLSET] 0 ENTER ENTER Use the zero feature. Graph, then select zero from the CALC menu. KEYSTROKES: GRAPH nd [CALC] [0, 0] scl: by [0, 0] scl: Place the cursor to the left of the zero and press ENTER for the Left Bound. Then place the cursor to the right of the zero and press ENTER for the Right Bound. Press ENTER to solve. The solution is about.6. This agrees with the estimate made by using the TABLE. 68 Chapter Polynomials

58 Instead of rewriting an equation or inequality so that one side is 0, you can also treat each side of the equation or inequality as a separate function and graph both. Solve. Graph each side of the inequality. In the Y= list, enter y and y. Then press GRAPH. [0, 0] scl: by [0, 0] scl: Use the trace feature. Press TRACE. You can use or to switch the cursor between the two curves. [0, 0] scl: by [0, 0] scl: The calculator screen above shows that, for points to the left of where the curves cross, Y Y or. To solve the original inequality, you must find points for which Y Y. These are the points to the right of where the curves cross. Assess When eamining the table of values for the equation on p. 68, why do you know a solution lies between the two values where the graphed function changes signs? Sample answer: Since the original equation was solved so that one side was zero, the function representing the other side has a value of 0 when the value of y is 0. That must occur for a value of somewhere between two values of whose corresponding values of y have different signs (because 0 lies between the positive numbers and the negative numbers). Use the intersect feature. You can use the INTERSECT feature on the CALC menu to approimate the -coordinate of the point at which the curves cross. KEYSTROKES: nd [CALC] Press ENTER for each of First curve?, Second curve?, and Guess?. The calculator screen shows that the -coordinate of the point at which the curves cross is about.0. Therefore, the solution of the inequality is about.0. Use the symbol instead of in the solution because the symbol in the original inequality is. [0, 0] scl: by [0, 0] scl: Answer 0. Rewrite the inequality so that one side is 0. Then graph the other side and find the values for which the graph is above or below the -ais, according to the inequality symbol. Use the zero feature to approimate the -coordinate of the point at which the graph crosses the -ais. Eercises. about.89. about. 8. about 0 9. about. Solve each equation or inequality Eplain how you could apply the technique in the first eample to solving an inequality. See margin. Graphing Calculator Investigation Solving Radical Equations and Inequalities by Graphing 69 Graphing Calculator Investigation Solving Radical Equations and Inequalities by Graphing 69

59 Lesson Notes Comple Numbers Focus -Minute Check Transparency -9 Use as a quiz or a review of Lesson -8. Mathematical Background notes are available for this lesson on p. 0D. do comple numbers apply to polynomial equations? Ask students: If by definition i, then what do you think is the value of i? Justify your answer. Since i (i ), replacing i with gives i () or. Teach ADD AND SUBTRACT COMPLEX NUMBERS In-Class Eamples Simplify. a. 8 i 7 b. y i y y Simplify. a. i i 6 b. 6 Simplify i. i Power Point Resource Manager Vocabulary imaginary unit pure imaginary number comple number absolute value comple conjugates Study Tip Reading Math i is usually written before radical symbols to make it clear that it is not under the radical. TEACHING TIP Point out that when multiplying radicals with negative radicands, students should first take the roots, then multiply. Otherwise, their answers may be off by a factor of. 70 Chapter Polynomials Add and subtract comple numbers. Multiply and divide comple numbers. Consider the equation 0. If you solve this equation for, the result is. Since there is no real number whose square is, the equation has no real solutions. French mathematician René Descartes (96 60) proposed that a number i be defined such that i. ADD AND SUBTRACT COMPLEX NUMBERS Since i is defined to have the property that i, the number i is the principal square root of ; that is, i. i is called the imaginary unit. Numbers of the form i, i, and i are called pure imaginary numbers. Pure imaginary numbers are square roots of negative real numbers. For any positive real number b, b b or bi. Eample Square Roots of Negative Numbers Simplify. a. 8 8 i or i Eample Multiply Pure Imaginary Numbers b. i or i The Commutative and Associative Properties of Multiplication hold true for pure imaginary numbers. Simplify. a. i 7i b. 0 i 7i i 0 i0 i () i i You can use the properties of powers to help simplify powers of i. Eample Simplify i. i i i i (i ) do comple numbers apply to polynomial equations? i () Simplify a Power of i Multiplying powers Power of a Power i i or i () Workbook and Reproducible Masters Chapter Resource Masters Study Guide and Intervention, pp Skills Practice, p. 89 Practice, p. 90 Reading to Learn Mathematics, p. 9 Enrichment, p. 9 Assessment, p. 08 School-to-Career Masters, p. 0 Teaching Algebra With Manipulatives Masters, pp. 9, 0 Transparencies -Minute Check Transparency -9 Answer Key Transparencies Technology Interactive Chalkboard

60 Study Tip Quadratic Solutions Quadratic equations always have comple solutions. If the discriminant is: negative, there are two imaginary roots, zero, there are two equal real roots, or positive, there are two unequal real roots. The solutions of some equations involve pure imaginary numbers. Eample Solve Equation with Imaginary Solutions Original equation Subtract 8 from each side. 6 Divide each side by. 6 Take the square root of each side. i 6 6 Consider an epression such as i. Since is a real number and i is a pure imaginary number, the terms are not like terms and cannot be combined. This type of epression is called a comple number. Words Eamples Comple Numbers A comple number is any number that can be written in the form a bi, where a and b are real numbers and i is the imaginary unit. a is called the real part, and b is called the imaginary part. 7 i and 6i (6)i In-Class Eamples Solve y 0 0. i Power Point Teaching Tip Make sure students understand that when they take the square root of both sides of an equation, they must use the symbol in front of the radical sign. Find the values of and y that make the equation yi i true. 7, y Teaching Tip Emphasize that two comple numbers are equal if and only if their real parts are equal and their imaginary parts are equal. The Venn diagram at the right shows the comple number system. If b 0, the comple number is a real number. If b 0, the comple number is imaginary. If a 0, the comple number is a pure imaginary number. Comple Numbers (a bi) Real Numbers b 0 Imaginary Numbers (b 0) Pure Imaginary Numbers a 0 Two comple numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a bi c di if and only if a c and b d. Study Tip Reading Math The form a bi is sometimes called the standard form of a comple number. Eample Equate Comple Numbers Find the values of and y that make the equation (y )i i true. Set the real parts equal to each other and the imaginary parts equal to each other. Real parts 6 y Add to each side. Divide each side by. y 6 Imaginary parts Add to each side. Lesson -9 Comple Numbers 7 Differentiated Instruction ELL Verbal/Linguistic Have students write poems about the imaginary number i and the repeating values of its powers, perhaps including wordplay with the terms real and imaginary. The content of the poems should be helpful for remembering the mathematical characteristics of i. Lesson -9 Comple Numbers 7

61 In-Class Eample 6 Simplify. a. ( i) ( i) i b. ( 6i) ( 7i) i Answers Algebra Activity. imaginary b ( i) ( i) O real a Power Point To add or subtract comple numbers, combine like terms. That is, combine the real parts and combine the imaginary parts. Eample 6 Add and Subtract Comple Numbers Simplify. a. (6 i) ( i) (6 i) ( i) (6 ) ( )i Commutative and Associative Properties 7 i Simplify. b. ( i) ( i) ( i) ( i) ( ) [ ()]i Commutative and Associative Properties i Simplify. You can model the addition and subtraction of comple numbers geometrically.. Rewrite the difference as a sum, ( i) ( i) ( i) ( i). Then apply the method discussed in this activity. Adding Comple Numbers You can model the addition of comple numbers on a coordinate plane. The horizontal ais represents the real part a of the comple number, and the vertical ais represents the imaginary part b of the comple number. Use a coordinate plane to find ( i ) ( i ). Create a coordinate plane and label the aes appropriately. Graph i by drawing a segment from the origin to (, ) on the coordinate plane. Represent the addition of i by moving units to the left and units up from (, ). You end at the point (, ), which represents the comple number i. So, ( i ) ( i ) i. b imaginary O i i real a Model and Analyze. Model ( i ) ( i ) on a coordinate plane. See margin.. Describe how you could model the difference ( i ) ( i ) on a coordinate plane. See margin.. The absolute value of a comple number is the distance from the origin to the point representing that comple number in a coordinate plane. Refer to the graph above. Find the absolute value of i. 9. Find an epression for the absolute value of a bi. a b MULTIPLY AND DIVIDE COMPLEX NUMBERS Comple numbers are used with electricity. In a circuit with alternating current, the voltage, current, and impedance, or hindrance to current, can be represented by comple numbers. 7 Chapter Polynomials 7 Chapter Polynomials Algebra Activity Materials: grid paper, ruler/straightedge The horizontal ais is often called a real number line. What might be a corresponding name for the vertical ais? an imaginary number line Where do real numbers lie on this coordinate plane? on the horizontal ais Where do pure imaginary numbers lie? on the vertical ais Where do comple numbers for which neither a nor b is 0 lie on this coordinate plane? on the regions of the plane other than the aes

62 Study Tip Reading Math Electrical engineers use j as the imaginary unit to avoid confusion with the I for current. Study Tip Concept Check Look Back Refer to Chapter to review the properties of fields and the properties of equality. Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 6, , 7, 8 You can use the FOIL method to multiply comple numbers. Eample 7 Multiply Comple Numbers ELECTRICITY In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E I Z. Find the voltage in a circuit with current j amps and impedance 7 j ohms. E I Z Electricity formula ( j) (7 j) (7) (j) (j)7 j(j) 7 j j j 7 6j () 6j The voltage is 6j volts. I j, Z 7 j FOIL Multiply. j Add. Two comple numbers of the form a bi and a bi are called comple conjugates. The product of comple conjugates is always a real number. For eample, ( i)( i) 6i 6i 9 or. You can use this fact to simplify the quotient of two comple numbers. Eample 8 Divide Comple Numbers Simplify. i a. b. i i i i i i i and i i i i i i i i are conjugates. i i 6 i i 6i Multiply. i i i 6i i 0 i Multiply. i i Standard form 0 i Standard form. Determine if each statement is true or false. If false, find a countereample. a. Every real number is a comple number. true b. Every imaginary number is a comple number. true. Decide which of the properties of a field and the properties of equality that the set of comple numbers satisfies. all of them. OPEN ENDED Write two comple numbers whose product is 0. Sample answer: i and i Simplify.. 6 6i. 0 y i y 6. (6i)(i) i 9 i 9. (8 6i) ( i) 6 i i 7 0. ( i)( 6i) i. i 7 7 i Why multiply by i i intead of i? i Lesson -9 Comple Numbers 7 MULTIPLY AND DIVIDE COMPLEX NUMBERS In-Class Eamples Practice/Apply Study Notebook Power Point 7 ELECTRICITY In an AC circuit, the voltage E, current I, and impedence Z are related by the formula E I Z. Find the voltage in a circuit with current j amps and impedence 6j ohms. 7 6j 8 Simplify. i 0 a. i i i b. i i Have students complete the definitions/eamples for the remaining terms on their Vocabulary Builder worksheets for Chapter. write a list of the first four powers of i: i, i, i i, i. include any other item(s) that they find helpful in mastering the skills in this lesson. Lesson -9 Comple Numbers 7

63 Study Study Guide and and Intervention Intervention, p. 87 (shown) and p NAME DATE PERIOD To solve a quadratic equation that does not have real solutions, you can use the fact that i to find comple solutions. Eample Solve 0. 0 Original equation Comple Numbers Add and Subtract Comple Numbers A comple number is any number that can be written in the form a bi, Comple Number where a and b are real numbers and i is the imaginary unit (i ). a is called the real part, and b is called the imaginary part. Addition and Combine like terms. Subtraction of (a bi) (c di) (a c) (b d )i Comple Numbers (a bi) (c di) (a c) (b d )i Eample Simplify (6 i) ( i). Eample Simplify (8 i) (6 i). (6 i) ( i) (8 i) (6 i) (6 ) ( )i (8 6) [ ()]i 0 i i Subtract from each side. Divide each side by. Eercises i Take the square root of each side. Simplify.. ( i) (6 i). ( i) ( i). (6 i) ( i) i i 0 i. ( i) ( i). (8 i) (8 i) 6. ( i) (6 i) 9i 6 i 7. ( i) ( i) 8. (9 i) ( i) 9. ( i) ( i) 8 8i 7 7i 6 i 0. i. i 6. i i Solve each equation i i6 i -9 Simplify.. 9 7i NAME 87 DATE PERIOD Practice (Average) Skills Practice, p. 89 and Practice, p. 90 (shown) Comple Numbers. 6 i. s 8 s i. 6a b a b ia 6 7. (i)(i)(i) 8. (7i) (6i) 9. i 60i 9i 0. i. i 89. ( i) ( 8i) i i 8 0i. (7 6i) (9 i). ( 8i) ( i). (0 i) (8 0i) 6 i 69i 8 i 6. (8 i) (0 0i) 7. (6 i)(6 i) 8. (8 i)(8 i) 8 6i 7 76i 9. ( i)( i) 0. (7 i)(9 6i). i 7 i 6i i 7 i i i i i i Solve each equation.. n 0 i7 6. m 0 0 i 7. m 76 0 i9 8. m 6 0 i 9. m 6 0 i 0. 0 i 6 i i 6i Find the values of m and n that make each equation true.. 8i m ni, 7. (6 m) ni 7i 8, 9. (m ) ( n)i 6 i, 6. (7 n) (m 0)i 6i,. ELECTRICITY The impedance in one part of a series circuit is j ohms and the impedance in another part of the circuit is 7 j ohms. Add these comple numbers to find the total impedance in the circuit. 8 j ohms 6. ELECTRICITY Using the formula E IZ, find the voltage E in a circuit when the current I is j amps and the impedance Z is j ohms. j volts Reading to Learn Mathematics, p. 9-9 NAME 90 DATE PERIOD Reading to Learn Mathematics Comple Numbers Pre-Activity How do comple numbers apply to polynomial equations? Read the introduction to Lesson -9 at the top of page 70 in your tetbook. Suppose the number i is defined such that i. Complete each equation. i (i) i Reading the Lesson. Complete each statement. a. The form a bi is called the standard form of a comple number. b. In the comple number i, the real part is and the imaginary part is. This is an eample of a comple number that is also a(n) imaginary number. c. In the comple number, the real part is and the imaginary part is 0. This is eample of comple number that is also a(n) real number. d. In the comple number 7i, the real part is 0 and the imaginary part is 7. This is an eample of a comple number that is also a(n) pure imaginary number.. Give the comple conjugate of each number. a. 7i b. i 7i i. Why are comple conjugates used in dividing comple numbers? The product of comple conjugates is always a real number.. Eplain how you would use comple conjugates to find ( 7i) ( i). Write the division in fraction form. Then multiply numerator and denominator by i. Helping You Remember. How can you use what you know about simplifying an epression such as to help you remember how to simplify fractions with imaginary numbers in the denominator? Sample answer: In both cases, you can multiply the numerator and denominator by the conjugate of the denominator. 7 Chapter Polynomials ELL Lesson -9 GUIDED PRACTICE KEY Eercises Eamples, 6 7 Chapter Polynomials NAME DATE PERIOD -9 Enrichment, p. 9 Conjugates and Absolute Value When studying comple numbers, it is often convenient to represent a comple number by a single variable. For eample, we might let z yi. We denote the conjugate of z by. z Thus, z yi. We can define the absolute value of a comple number as follows. z yi y There are many important relationships involving conjugates and absolute values of comple numbers. Eample Show z zz for any comple number z. Let z yi. Then, z ( yi)( yi) y ( ) y z Application indicates increased difficulty Practice and Apply Homework Help For Eercises See Eamples , 6, 6 7 7,, 7 8,, Etra Practice See page (i ) ( i) i More About... Electrical Engineering The chips and circuits in computers are designed by electrical engineers. Online Research To learn more about electrical engineering, visit: com/careers Solve each equation i. 0 i. 0 i Find the values of m and n that make each equation true.. m (n )i 6 8i, 6. (n ) (m )i 7i, 7. ELECTRICITY The current in one part of a series circuit is j amps. The current in another part of the circuit is 6 j amps. Add these comple numbers to find the total current in the circuit. 0 j amps Simplify i. i. 6 7i 8. i i i. 00a b 0a b i (i)(6i)(i) 8i. i(i) 7i 6. i i 7. i 8. i 8 9. i 6 i 0. ( i) ( i). ( i) ( i) 6. ( i) ( i). (7 i) ( i). ( i)( i). ( i)( i) i 6. (6 i)( i) 8 i 7. ( i)( i) 8 i 8. i 6 i i 7 7 i 0. 0 i i i. i i i. ( i)(6 i)( i) 6 6i. ( i)( i)( i) 0 i. i i. i i i i 6. Find the sum of i ( i) and ( i) i. 7. Simplify [( i) i i] [( i) ( i) ]. ( i) ( i) 7 i Solve each equation i i 0. 0 i i. 9 0 i i i. 0 i Find the values of m and n that make each equation true , i m ni, 7. (m ) ni 9i, 8. (m ) ( n)i i 7, 9. ( n) (m 7)i 8 i, 60. (m n) (m n)i i, 6. (m n)i (m n) 7i 6. ELECTRICITY The impedance in one part of a series circuit is j ohms, and the impedance in another part of the circuit is 6j. Add these comple numbers to find the total impedance in the circuit. j ohms ELECTRICAL ENGINEERING For Eercises 6 and 6, use the formula E I Z. 6. The current in a circuit is j amps, and the impedance is j ohms. What is the voltage? 8j volts 6. The voltage in a circuit is 8j volts, and the impedance is j ohms. What is the current? j amps

64 Standardized Test Practice Etending the Lesson Maintain Your Skills Mied Review CRITICAL THINKING Show that the order relation does not make sense for the set of comple numbers. (Hint: Consider the two cases i 0 and i 0. In each case, multiply each side by i.) See pp. 8A 8B. 66. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How do comple numbers apply to polynomial equations? Include the following in your answer: how the a and c must be related if the equation a c 0 has comple solutions, and the solutions of the equation If i, then what is the value of i 7? C A B 0 C i D i 68. The area of the square is 6 square units. What is the area of the circle? C A units B units C units D 6 units PATTERN OF POWERS OF i 69., i,, i,, i,, i, 69. Find the simplified forms of i 6, i 7, i 8, i 9, i 0, i, i, i, and i. 70. Eplain how to use the eponent to determine the simplified form of any power of i. See margin. Solve each equation. (Lesson -8) Simplify each epression. (Lesson -7) y y 76. a For Eercises 77 80, triangle ABC is reflected over the -ais. (Lesson -6) 77. Write a verte matri for the triangle. 78. Write the reflection matri. 79. Write the verte matri for A B C. 80. Graph A B C. See pp. 8A 8B. 8. FURNITURE A new sofa, love seat, and coffee table cost $00. The sofa costs twice as much as the love seat. The sofa and the coffee table together cost $0. How much does each piece of furniture cost? (Lesson -) sofa: $00, love seat: $600, coffee table: $0 Graph each system of inequalities. (Lesson -) 8 8. See pp. 8A 8B. 8. y 8. y y y C a a O y B A About the Eercises Organization by Objective Add and Subtract Comple Numbers: 8, 6 6 Multiply and Divide Comple Numbers: Odd/Even Assignments Eercises 8 6 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 9 odd, 7 9 odd, 6, 6 68, 7 8 Average: 9 6 odd, 6 68, 7 8 (optional: 69, 70) Advanced: 8 6 even, 6 8 Assess Open-Ended Assessment Speaking Have students discuss the meaning and reality of imaginary numbers, including their graphical representation and their usefulness in electrical engineering. Intervention Suggest that students who are confused by imaginary numbers think of i as a very special kind of variable that most of the time can be treated similar to the variable. New Find the slope of the line that passes through each pair of points. (Lesson -) 8. (, ), (8, ) 8. (, ), (, ) Answers 66. Some polynomial equations have comple solutions. Answers should include the following. a and c must have the same sign. i Lesson -9 Comple Numbers Eamine the remainder when the eponent is divided by. If the remainder is 0, the result is. If the remainder is, the result is i. If the remainder is, the result is. And if the remainder is, the result is i. Assessment Options Quiz (Lessons -8 and -9) is available on p. 08 of the Chapter Resource Masters. Lesson -9 Comple Numbers 7

65 Study Guide and Review Vocabulary and Concept Check This alphabetical list of vocabulary terms in Chapter includes a page reference where each term was introduced. Assessment A vocabulary test/review for Chapter is available on p. 06 of the Chapter Resource Masters. Lesson-by-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabulary PuzzleMaker ELL The Vocabulary PuzzleMaker software improves students mathematics vocabulary using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts ( questions) Round Skills ( questions) Round Problem Solving ( questions) Vocabulary and Concept Check absolute value (p. 7) binomial (p. 9) coefficient (p. ) comple conjugates (p. 7) comple number (p. 7) conjugates (p. ) constant (p. ) degree (p. ) - See pages Chapter Polynomials dimensional analysis (p. ) etraneous solution (p. 6) FOIL method (p. 0) imaginary unit (p. 70) like radical epressions (p. ) like terms (p. 9) monomial (p. ) nth root (p. ) polynomial (p. 9) power (p. ) principal root (p. 6) pure imaginary number (p. 70) radical equation (p. 6) radical inequality (p. 6) rationalizing the denominator (p. ) scientific notation (p. ) simplify (p. ) square root (p. ) standard notation (p. ) synthetic division (p. ) terms (p. 9) trinomial (p. 9) Choose a word or term from the list above that best completes each statement or phrase.. A number is epressed in when it is in the form a 0 n, where a 0 and n is an integer. scientific notation. A shortcut method known as is used to divide polynomials by binomials. synthetic division. The is used to multiply two binomials. FOIL method. A(n) is an epression that is a number, a variable, or the product of a number and one or more variables. monomial. A solution of a transformed equation that is not a solution of the original equation is a(n). etraneous solution 6. are imaginary numbers of the form a bi and a bi. Comple conjugates 7. For any number a and b, if a b, then a is a(n) of b. square root 8. A polynomial with three terms is known as a(n). trinomial 9. When a number has more than one real root, the is the nonnegative root. principal root 0. i is called the. imaginary unit Eamples For more information about Foldables, see Teaching Mathematics with Foldables. TM Monomials Concept Summary The properties of powers for real numbers a and b and integers m and n are as follows. a n a n, a 0 (a m ) n a mn a m a n a m n (ab) m a m b m a m an a m n, a 0 a b a, n b n b 0 Use scientific notation to represent very large or very small numbers. Simplify ( y 6 )(8 y). ( y 6 )(8 y) ()(8) y 6 Commutative Property and products of powers 7 y 7 Simplify. Ask students to review their Foldable and make sure that their notes, diagrams, and eamples are complete. Since journal entries are personal, remind students that these journals are shared only with their consent. Ask if anyone would like to describe one of their journal entries, perhaps something they had difficulty with but later cleared up by asking questions. Encourage students to refer to their Foldables while completing the Study Guide and Review and to use them in preparing for the Chapter Test. 76 Chapter Polynomials

66 Chapter Study Guide and Review Study Guide and Review - See pages 9. Eamples Eample Epress each number in scientific notation. a.,000 b ,000. 0, , or Eercises Simplify. Assume that no variable equals 0. See Eamples on pages.. f 7 f f. ( ) 7 6. (y)(y ) 8y. c f cd Evaluate. Epress the result in scientific notation. See Eamples 7 on page.. (000)(8,000) (0.00) , 00, Polynomials Concept Summary Add or subtract polynomials by combining like terms. Multiply polynomials by using the Distributive Property. Multiply binomials by using the FOIL method. - Dividing Polynomials See pages Concept Summary 8. Use the division algorithm or synthetic division to divide polynomials. Use synthetic division to find ( 9 ) ( ) The quotient is Eercises Simplify. See Eamples on pages... ( 6 ) ( ) 6. (0 9) ( ) 7. ( ) ( ) 8. ( 8 0 ) ( ) 6 c d f Simplify ( ) ( 6 7). Find (9k )(7k 6). ( 6 7) (9k )(7k 6) 6 7 (9k)(7k) (9k)(6) ()(7k) ()(6) ( ) ( 6) 7 6k k 8k 7 6k 6k 8. c m n 78m 0m n Eercises Simplify. See Eamples on pages 9 and (c ) (c ) (6c 7) 9. ( ) (7 9 9) 0. 6m (mn m n). 8 y 0 ( y 9 0 y 6 ) y y. (d )(d ). (a 6). (b c) d d a a 6 8b 6b c bc 7c Chapter Study Guide and Review 77 Chapter Study Guide and Review 77

67 Study Guide and Review Chapter Study Guide and Review - See pages 9. Eamples Factoring Polynomials Concept Summary You can factor polynomials using the GCF, grouping, or formulas involving squares and cubes. Factor ( 6 ) (0 ) ( ) ( ) ( )( ) Group to find the GCF. Factor the GCF of each binomial. Distributive Property Factor m m. Find two numbers whose product is () or, and whose sum is. The two numbers must be and because () and (). m m m m m (m m) (m ) m(m ) ()(m ) (m )(m ) Eercises Factor completely. If the polynomial is not factorable, write prime. See Eamples on pages 9 and. 0. (a )(a ). (w )(w ) ( )( ) 0. 0a 0a a. w 0w w. 7 ( )( ). s (s 8)(s 8s 6). 7 prime - See pages 9. Eamples Roots of Real Numbers Concept Summary Real nth roots of b, n b, or n b n n b if b 0 n b if b 0 n b if b 0 even odd one positive root one negative root one positive root no negative roots no real roots no positive roots one negative root one real root, 0 Simplify 8. 6 Simplify y 8 6 (9 ) 8 6 = (9 ) 7 87 y 7 ( ) y 7 87 y ( y ) 7 9 Use absolute value. y Evaluate. Eercises Simplify. See Eamples and on pages 6 and (8) 8 8. c d cd 9. ( ) 0. ( ). 6m 8 m. a0a a 78 Chapter Polynomials 78 Chapter Polynomials

68 Chapter Study Guide and Review Study Guide and Review -6 See pages 0 6. Eample Radical Epressions Concept Summary For any real numbers a and b and any integer n, Product Property: n ab n a n b Quotient Property: n a n a b n b Simplify 6 m 0m. 6 m 0m 6 (m m 0 ) 0 m 0 m 0 m or 0m Product Property of Radicals Factor into eponents of if possible. Product Property of Radicals Write the fifth roots. Eercises Simplify. See Eamples 6 on pages Radical Eponents See pages Concept Summary 7 6. For any nonzero real number b, and any integers m and n, with n, b m n n b m n b m Eamples Write in radical form. Simplify. z Product of powers Rational eponents z z 6 Add. ( ) 6 6 or 6 Power of a power z z z z z or z z Rationalize the denominator. Rewrite in radical form. Eercises Evaluate. See Eamples and on pages 8 and Simplify. See Eample on page 9. y y y yz z z y Chapter Study Guide and Review 79 Chapter Study Guide and Review 79

69 Study Guide and Review Etra Practice, see pages Mied Problem Solving, see page See pages Eample Radical Equations and Inequalities Concept Summary To solve a radical equation, isolate the radical. Then raise each side of the equation to a power equal to the inde of the radical. Solve Original equation Subtract from each side. Square each side. Evaluate the squares. Solve for. Eercises Solve each equation. See Eamples on pages 6 and y ( ) 8 no solution t y y y y 8-9 See pages Eamples Comple Numbers Concept Summary i and i Comple conjugates can be used to simplify quotients of comple numbers. Simplify ( i) ( i). ( i) ( i) [ ()] ( )i Group the real and imaginary parts. i Add. 7i Simplify. i 7i 7i i i i i i i 9i i or i i and i are conjugates. Multiply. i Eercises Simplify. See Eamples and 6 8 on pages 70, 7, and i 67. 6m 8m 6 i 68. (7 i) ( 6i) i 6 7. ( i)( i) i 7. 6 i6 i 7 7. i i 7. i i i i 7. 9i i i 0 80 Chapter Polynomials 80 Chapter Polynomials

70 Practice Test Vocabulary and Concepts Choose the term that best describes the shaded part of each trinomial.. c a. degree. 6 a. 9 7 b Skills and Applications Simplify. 6. 8h 7h 6h 6. (b) (6c),00b c. ( )( ) 8 6. (h 6) Evaluate. Epress the result in scientific notation. 7. (.6 0 )( 0 ) ,00, Simplify. 9. ( 0 ) ( ) 0. ( 9 7) ( ) 8 Factor completely. If the polynomial is not factorable, write prime.. ( )( 9). r pr p (r p)(r p). Simplify s t 8 s t 6 s 7. v b v. v b b b Solve each equation.. b b 7. no solution. y w Simplify. 9. ( i) (8 i) 9i 0. ( i) 7 0i. SKYDIVING The approimate time t in seconds that it takes an object to fall d a distance of d feet is given by t. Suppose a parachutist falls seconds 6 before the parachute opens. How far does the parachutist fall during this time period? 96 ft. GEOMETRY The area of a triangle with sides of length a, b, and c is given by s(s )(s a)(s b), c where s (a b c). If the lengths of the sides of a triangle are 6, 9, and feet, what is the area of the triangle epressed in radical form? 7 ft. STANDARDIZED TEST PRACTICE D A B C D b. constant term c. coefficient Chapter Practice Test 8 Assessment Options Vocabulary Test A vocabulary test/review for Chapter can be found on p. 06 of the Chapter Resource Masters. Chapter Tests There are si Chapter Tests and an Open- Ended Assessment task available in the Chapter Resource Masters. Chapter Tests Form Type Level Pages MC basic 9 9 A MC average 9 96 B MC average C FR average D FR average 0 0 FR advanced 0 0 MC = multiple-choice questions FR = free-response questions Open-Ended Assessment Performance tasks for Chapter can be found on p. 0 of the Chapter Resource Masters. A sample scoring rubric for these tasks appears on p. A. TestCheck and Worksheet Builder This networkable software has three modules for assessment. Worksheet Builder to make worksheets and tests. Student Module to take tests on-screen. Management System to keep student records. Portfolio Suggestion Introduction In this chapter, you have divided and simplified monomials, polynomials, radical epressions, and comple numbers, often using procedures that involved a series of steps. Ask Students Write a description for your portfolio comparing these various division problems. Identify which type of division problems was most challenging for you and eplain why you think this is true. Be sure to include several eamples of your work from this chapter. Chapter Practice Test 8

71 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequently given standardized tests. A practice answer sheet for these two pages can be found on p. A of the Chapter Resource Masters. Standardized Test Practice NAME DATE PERIOD Standardized Test Practice Student Recording Sheet (Use with pages Sheet, 8 8 ofp. the Student AEdition.) Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 7 A B C D A B C D A B C D 8 A B C D A B C D 6 A B C D 9 A B C D Part Short Response/Grid In Solve the problem and write your answer in the blank. Also enter your answer by writing each number or symbol in a bo. Then fill in the corresponding oval for that number or symbol. 0 6 / / / / / / / / Part Quantitative Comparison / / / / / / / / Select the best answer from the choices given and fill in the corresponding oval. 8 A B C D 0 A B C D A B C D 9 A B C D A B C D Additional Practice See pp. in the Chapter Resource Masters for additional standardized test practice. Answers Part Multiple Choice Record your answers on the answer sheet provided by your teacher or on a sheet of paper.. If 0 and is a real number, then lies between which two consecutive integers? B A and B and C and D and 6. If 7y 9 and y, then what is the value of 8 8y? C A B 8 C 6 D. For all positive integers n, n n n, if n is even and (n ), if n is odd. What is 8? B A B C D Let y y y for all integers and y. If y 0 and y 0, what must equal? D A B C 0 D. The sum of a number and its square is three times the number. What is the number? D A 0 only B only C only D 0 or 6. In rectangle ABCD, AD is 8 units long. What is the length of AB in units? C A B C D The sum of two positive consecutive integers is s. In terms of s, what is the value of the greater integer? D s s A B s s C D 8. Latha, Renee, and Cindy scored a total of 0 goals for their soccer team this season. Latha scored three times as many goals as Renee. The combined number of goals scored by Latha and Cindy is four times the number scored by Renee. How many goals did Latha score? C A C B 6 8 D 0 9. If s t and t, then which of the following must be equal to s t? D A C D (s t) B t s D s t A Test-Taking Tip 0 Question 9 If you simplify an epression and do not find your answer among the given answer choices, follow these steps. First, check your answer. Then, compare your answer with each of the given answer choices to determine whether it is equivalent to any of the answer choices. B C 8 Chapter Standardized Test Practice Log On for Test Practice The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit or TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra End-of-Course tests can be found on this CD-ROM. 8 Chapter Polynomials

72 Aligned and verified by Part : QUANTITATIVE COMPARISON Part Short Response/Grid In Part Quantitative Comparison Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 0. Let a b a, where b 0. What is the b value of?. or /. If 7, what is the value of?. In the figure, if and z 0, what is the value of y? 0 z y. For all positive integers n, let n equal the greatest prime number that is a divisor of n. What does 70 equal? 7/ 7. If y 6 and y, then?.. In the figure, a square with side of length is inscribed in a circle. If the area of the circle is k, what is the eact value of k? Compare the quantity in Column A and the quantity in Column B. Then determine whether: A B C D the quantity in Column A is greater, the quantity in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given. 8. s and t are positive integers. A 9. The original price of a VCR is discounted by 0%, giving a sale price of $08. the original price $0 of the VCR A 0. w Column A s t s w Column B s s t w B the area of the rectangle the area of the circle 6. For all nonnegative numbers n, let n be defined by n. n If n, what is the value of n? 6 7. For the numbers a, b, and c, the average (arithmetic mean) is twice the median. If a 0, and a b c, what is the value of b c? k and n are integers. k n 6 D. For all positive integers m and p, let m p (m p) mp. C k 8 8 n Chapter Standardized Test Practice 8 Chapter Standardized Test Practice 8