Chapter 5 Resource Masters

Transcription

1 Chapter Resource Masters

2 Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Study Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook ANSWERS FOR WORKBOOKS The answers for Chapter of these workbooks can be found in the back of this Chapter Resource Masters booklet. Glenc oe/ McGraw-Hill Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe s Algebra. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill Companies Orion Place Columbus, OH 0-0 ISBN: Algebra Chapter Resource Masters

3 Contents Vocabulary Builder vii Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson -9 Study Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Chapter Assessment Chapter Test, Form Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Test, Form D Chapter Test, Form Chapter Open-Ended Assessment Chapter Vocabulary Test/Review Chapter Quizzes & Chapter Quizzes & Chapter Mid-Chapter Test Chapter Cumulative Review Chapter Standardized Test Practice.. Standardized Test Practice Student Recording Sheet A ANSWERS A A Glencoe/McGraw-Hill iii Glencoe Algebra

4 Teacher s Guide to Using the Chapter Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter Resource Masters includes the core materials needed for Chapter. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra TeacherWorks CD-ROM. Vocabulary Builder Pages vii viii include a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. WHEN TO USE Give these pages to students before beginning Lesson -. Encourage them to add these pages to their Algebra Study Notebook. Remind them to add definitions and examples as they complete each lesson. Study Guide and Intervention Each lesson in Algebra addresses two objectives. There is one Study Guide and Intervention master for each objective. WHEN TO USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN TO USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closely follow the structure of the Practice and Apply section of the Student Edition exercises. These exercises are of average difficulty. WHEN TO USE These provide additional practice options or may be used as homework for second day teaching of the lesson. Reading to Learn Mathematics One master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the context of and relationships among terms in the lesson. Finally, students are asked to summarize what they have learned using various representation techniques. WHEN TO USE This master can be used as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one extension master for each lesson. These activities may extend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students. WHEN TO USE These may be used as extra credit, short-term projects, or as activities for days when class periods are shortened. Glencoe/McGraw-Hill iv Glencoe Algebra

5 Assessment Options The assessment masters in the Chapter Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Chapter Assessment CHAPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms A and B contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms C and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with axes are provided for questions assessing graphing skills. Form is an advanced level test with free-response questions. Grids without axes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. The Open-Ended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. A Vocabulary Test, suitable for all students, includes a list of the vocabulary words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate Assessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. Continuing Assessment The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of Algebra. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which may appear on the standardized tests that they may encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. Bubble-in and grid-in answer sections are provided on the master. Answers Page A is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages. This improves students familiarity with the answer formats they may encounter in test taking. The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer keys are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe Algebra

6 Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter. binomial Vocabulary Term Found on Page Definition/Description/Example Vocabulary Builder coefficient KOH uh FIH shuhnt complex conjugates KAHN jih guht complex number degree extraneous solution ehk STRAY nee uhs FOIL method imaginary unit like radical expressions like terms (continued on the next page) Glencoe/McGraw-Hill vii Glencoe Algebra

7 Reading to Learn Mathematics Vocabulary Builder (continued) monomial Vocabulary Term Found on Page Definition/Description/Example nth root polynomial power principal root pure imaginary number radical equation radical inequality rationalizing the denominator synthetic division sihn THEH tihk trinomial Glencoe/McGraw-Hill viii Glencoe Algebra

8 - Study Guide and Intervention 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 Exponent a n and a n for any real number a 0 and any integer n. a n a n When you simplify an expression, you rewrite it without parentheses or negative exponents. The following properties are useful when simplifying expressions. 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 or b n, a 0, b 0 a n b a Lesson - Example a. (m n )(mn) Simplify. Assume that no variable equals 0. (m n )(mn) m n m n m m n n m n m Exercises Simplify. Assume that no variable equals 0. b b. (m ) (m ) (m ) (m ) m m m m m. c c c c. b. (a ) a 0 b x y x y y.. a b x a b b a. x y xy x y. (a b ) (abc) a b c. m m m 9. m n mn m n 0. c t. j(j k )(j k j mn ). (m n) m c t k m n Glencoe/McGraw-Hill 9 Glencoe Algebra

9 - Study Guide and Intervention (continued) Monomials Scientific Notation Scientific notation A number expressed in the form a 0 n, where a 0 and n is an integer Example Express,000,000 in scientific notation.,000,000. 0,000, Write 0,000,000 as a power of ten. Example. 0 0 Exercises Evaluate. Express the result in scientific notation Express each number in scientific notation.., ,0, ,000, , , Evaluate. Express the result in scientific notation. 0. (. 0 )( 0 ). (. 0 )( 0 ). (. 0 )( 0 ) (. 0 ). (. 0 ). (. 0 ) ASTRONOMY Pluto is,. million miles from the sun. Write this number in scientific notation. Source: New York Times Almanac. 0 9 miles 0. CHEMISTRY The boiling point of the metal tungsten is 0,0 F. Write this temperature in scientific notation. Source: New York Times Almanac.0 0. BIOLOGY The human body contains 0.000% iodine by weight. How many pounds of iodine are there in a 0-pound teenager? Express your answer in scientific notation. Source: Universal Almanac. 0 lb Glencoe/McGraw-Hill 0 Glencoe Algebra

10 - Skills Practice Monomials Simplify. Assume that no variable equals 0.. b b b. c c c c 9. a a. x x x x a. (g ) g. (u) u. (x) x. (z) 0z 9. (d) d 0. (t ) t Lesson -. (r ) r. s s s k. 9. (f g) f 9 g k 0 k. (x) (y) x y. gh( g h ) g h. 0x y (0xy ) 00x y. wz w z z w a 9. bc c 0pq 0. r a b c a b p q r q p Express each number in scientific notation.., ,00, Evaluate. Express the result in scientific notation ( 0 )(. 0 ) Glencoe/McGraw-Hill Glencoe Algebra

11 - Simplify. Assume that no variable equals 0.. n n n. y y y y. t 9 t t. x x x. (f ) f. (b c ) c 9 b. (d t v )(dt v 0d ) t. u(z) uz v m 9. y m y s 0. x s 9my sx x x. (x ) x. x r s z 9r s z. (w z )(w). (m n ) (m n p ) m n p 0 wz Practice (Average) Monomials. d f d f d f 9. (x. y )(xy ) x 0(m v. v)(v) (x ) y y (v) (m ) m Express each number in scientific notation. x y x y y x 9. 9, x Evaluate. Express the result in scientific notation.. (. 0 )(.9 0 ). (. 0 9 )(. 0 ) COMPUTING The term bit, short for binary digit, was first used in 9 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. 0 miles per second, at what velocity does it travel through water? Write your answer in scientific notation..9 0 mi/s. 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 0 meters and a blue wavelength measures about. 0 meters, about how many times longer is the infrared wavelength than the blue wavelength? about. times Glencoe/McGraw-Hill Glencoe Algebra

12 - Reading to Learn Mathematics Monomials Pre-Activity Why is scientific notation useful in economics? Reading the Lesson Read the introduction to Lesson - at the top of page in your textbook. Your textbook gives the U.S. public debt as an example from economics that involves large numbers that are difficult to work with when written in standard notation. Give an example 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. Tell whether each expression is a monomial or not a monomial. If it is a monomial, tell whether it is a constant or not a constant. Lesson - a. x monomial; not a constant b. y y not a monomial c. monomial; constant d. not a monomial z. Complete the following definitions of a negative exponent and a zero exponent. 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 exponents that you would use to simplify each expression. (Do not actually simplify.) m a. quotient of powers m b. y 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 exponents and when to use negative ones. What is an easy way to remember this? Sample answer: Use a positive exponent if the number is 0 or greater. Use a negative number if the number is less than. Glencoe/McGraw-Hill Glencoe Algebra

13 - Enrichment Properties of Exponents The rules about powers and exponents are usually given with letters such as m, n, and k to represent exponents. For example, one rule states that a m a n a m n. In practice, such exponents are handled as algebraic expressions and the rules of algebra apply. Example 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 exponent n as n. It is important always to collect like terms only. Example Simplify (a n b m ). (a n b m ) (a n b m )(a n b m ) F O I L a n a n a n b m a n b m b m b m a n a n b m b m The second and third terms are like terms. Simplify each expression by performing the indicated operations.. m. (a ) n. ( n b ) k. (x a j ) m. (ay n ). (b k x). (c ) hk. (d n ) 9. (a b)(a n b ) 0. (x n y m )(x m y n ).. a n x x n. (ab a b)(a n b n ). ab (a b n ab n b n ) Glencoe/McGraw-Hill Glencoe Algebra

14 - Study Guide and Intervention 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. Example Simplify rs r s r rs s. rs r s r rs s (r r ) (rs rs) (s s ) r rs s Group like terms. Combine like terms. Example Simplify xy xy x y (0xy xy x y). xy xy x y (0xy xy x y) xy xy x y 0xy xy x y (x y x y ) (xy xy ) (xy 0xy) x y xy xy Exercises Simplify. Distribute the minus sign. Group like terms. Combine like terms. Lesson -. (x x ) (x x ). (y xy x ) (xy y x ) x x y xy x. (m m) (m m ). x y y x m m x y. (p pq q ) (p pq q ). j k j j k p pq 0q j k. (m n ) (n m ). bc b c b c bc 0m n b bc c 9. r s rs r s rs r s 9rs 0. 9xy x y (y xy x ) r s rs x xy 0y. (xy x y) (x y xy). 0.b.b. (.9b.b.) x xy y.9b.b 0.. (bc 9b c ) (c b bc). x y xy y xy 0x 0b bc c x xy y. x xy y xy y x. p p p p p p x xy y 9p p p Glencoe/McGraw-Hill Glencoe Algebra

15 - Study Guide and Intervention (continued) Polynomials Multiply Polynomials You use the distributive property when you multiply polynomials. When multiplying binomials, the FOIL pattern is helpful. FOIL Pattern Example To multiply two binomials, add the products of F the first terms, O the outer terms, I the inner terms, and L the last terms. Find y( y y ). y( y y ) y() y(y) y(y ) Distributive Property y y 0y Multiply the monomials. Example Find (x )(x ). (x )(x ) x x x () x () First terms Outer terms Inner terms Last terms x x 0x Multiply monomials. x x Add like terms. Exercises Find each product.. x(x ). a( a a ). y ( y y ) x 0x a a a y 0y y. (x )(x ). ( x)( x). (x )(x ) x x x x x x. (x )(x ). (x )(x ) 9. (x )(x 0) x x x x x x 0 0. (a c) (a c). (a )(a ). x(x ) x ( x) a c a 0a x x 0x. (t )(t ). (r ). (c )(c ) t t 0 r r 9 c c. (a )(a ). (x )(x x) a 9 x x x x. (x )(x ) 9. (x )(x x ) x x 0 x x x 0. (n )(n n ). (x )(x x ) n 0n n n x x x Glencoe/McGraw-Hill Glencoe Algebra

16 - Skills Practice Polynomials Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. b c d. x x yes;. no. xz y yes; Simplify.. (g ) (g ). (d ) (d ) g d. (x x ) (x x ). (f f ) (f f ) x x f f. (r r ) (r r ) 9. (x xy) (x xy y ) r 9r x xy y 0. (t ) (t t ). (u ) ( u u) t t u u 0 Lesson -. (c d ). x (x 9) 0c d x 9x. q(pq q ). w(hk 0h m k w ) pq q hk w 0h m w k w. m n (m n mnp ). s y(s y sy ) m n m n p m n s y 9s y s y. (c )(c ) 9. (z )(z ) c 0c z z 0. (a ). (x )(x ) a 0a x 9x. (r s)(r s). (y )(y ) r s y y. ( b)( b). (w ) 9 b 9w w Glencoe/McGraw-Hill Glencoe Algebra

17 - Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.. x xy xy yes;. ac a d yes;. no. x z x yes;. c c no. no Simplify. Practice (Average) Polynomials. (n ) (n ). (w w ) ( w ) n w w 9. (n n ) (n 9n ) 0. (x x) (x x ) 9n n x x. (m mp p ) (m mp p ). (x xy y ) (x xy y ) m mp p x xy y. (t ) (t t ). (u ) ( u u) t t u u 0. 9( y w). 9r y (ry r y r 0 ) 9y w r y 9 r y r y. a w(a w aw ). a w (a w a w 9aw ) a w a w a w 9 a w a w 9 9. x (x xy y ) 0. ab d (ab d ab) x x y x y a b d a b d. (v )(v ). (a 9y)(a y) v v a ay 9y. ( y ). (x y) y y x 0x y y. (x w)(x w). (n )(n ) x w n 9 m n 9 (m n). (w s)(w ws s ). (x y)(x xy y ) w s x x y xy y 9. BANKING Terry invests $00 in two mutual funds. The first year, one fund grows.% and the other grows %. Write a polynomial to represent the amount Terry s $00 grows to in that year if x represents the amount he invested in the fund with the lesser growth rate. 0.0x GEOMETRY The area of the base of a rectangular box measures x x square units. The height of the box measures x units. Find a polynomial expression for the volume of the box. x x x units Glencoe/McGraw-Hill Glencoe Algebra r s

18 - 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 textbook. Suppose that Shenequa decides to enroll in a five-year engineering program rather than a four-year program. Using the model given in your textbook, 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 $,0 by.0. Reading the Lesson. State whether the terms in each of the following pairs are like terms or unlike terms. a. x,y unlike terms b. m,m like terms c. r,s unlike terms d., like terms. State whether each of the following expressions is a monomial, binomial, trinomial, or not a polynomial. If the expression is a polynomial, give its degree. a. r r trinomial; degree b. x not a polynomial c. x y binomial; degree d. ab ab ab trinomial; degree Lesson -. 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 (x )(x ). Show the terms you would multiply, but do not actually multiply them. F (x)(x) O (x)() I ()(x) L ()() Helping You Remember. You can remember the difference between monomials, binomials, and trinomials by thinking of common English words that begin with the same prefixes. 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 Glencoe/McGraw-Hill 9 Glencoe Algebra

19 - Enrichment 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 p n p m n. x y z x y z x y z. a ab b a ab b. a ab b a ab b. a ab b a b. a a a a a. x x x x. x x x x x Glencoe/McGraw-Hill 0 Glencoe Algebra

20 - Study Guide and Intervention 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. Example p t r p qtr 9p tr p tr p Simplify t r p qtr 9p tr. p tr p t r p qtr 9p tr p tr p tr p tr p t r p qt r 9 p t r pt qr p Example Use long division to find (x x x 9) (x ). x x x x x x 9 ()x x x x ()x x x 9 ()x The quotient is x x, and the remainder is. x Therefore x x 9 x x. x x Exercises Simplify. a. 0a mn 0a. 0m n. b b a b a m n ab Lesson - a n b 0a 0 ab a m a. (x x ) (x ). (m m ) (m ) x m. (p ) (p ). (t t ) (t ) p p p t t. (x ) (x ) 9. (x x x ) (x ) x x x x x x m t Glencoe/McGraw-Hill Glencoe Algebra

21 - Study Guide and Intervention (continued) Dividing Polynomials Use Synthetic Division Synthetic division a procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x r Use synthetic division to find (x x x ) (x ). Step Write the terms of the dividend so that the degrees of the terms are in x x x descending order. Then write just the coefficients. Step Write the constant r of the divisor x r to the left, In this case, r. Bring down the first coefficient,, as shown. Step Multiply the first coefficient by r,. Write their product under the second coefficient. Then add the product and the second coefficient:. Step Multiply the sum,, by r:. Write the product under the next coefficient and add: (). Step Multiply the sum,, by r:. Write the product under the next coefficient and add: 0. The remainder is 0. 0 Thus, (x x x ) (x ) x x. Exercises Simplify.. (x x 9x ) (x ). (x x x ) (x ) x x x x. (x x 0x ) (x ). (x x 9x 9) (x ) x x x x x. (x 0x 9x ) (x ). (x x x ) (x ) x 9 x 0 x x x. (x 9x x ) (x ). (x x x 0) (x ) x x x x x x 9. (x x x 9) (x ) 0. (x x x x ) (x ) x x x x x x. (x 0x x 0x ) (x ) x x 0 x Glencoe/McGraw-Hill Glencoe Algebra

22 - Simplify. 0c Skills Practice Dividing Polynomials x 0. c. x y y y y x x x. y y. x x. (q q )(q ). (f f f f )(f ) q f f f q. (j k 9jk ) jk. (a h a h a ) (a ) j k h a ah 9. (n n 0) (n ) 0. (d d ) (d ) n d. (s s ) (s ). (y y )(y ) s y. (g 9) (g ). (x x ) (x ) g x x u. u. u u u x x x x x Lesson -. (v v 0)(v ). (t t t t 0)(t ) 0 v v t t y 9. y 0. y y y y x x 9x x x x x. (p p p) ( p ). (c c c )(c ) p p c p c. GEOMETRY The area of a rectangle is x x x square units. The width of the rectangle is x units. What is the length of the rectangle? x x units Glencoe/McGraw-Hill Glencoe Algebra

23 - Practice (Average) Dividing Polynomials Simplify. r. 0 r 0r r r k k. m k m 9m k r r km m 9m k. (0x y x y x y) (x y). (w z w z w z) (w z) x y wz z wz w. (a a a )(a). (d k d k dk )(dk ) a a a d d f f 0 f x x x. f. x 9. (a ) (a ) a a 0. (b ) (b ) b b 9 x x x x x x. x x. x x x. (w w w ) (w ). (y y y ) (y ) w w y w y y y. (x x x x 0) (x ). (m m ) (m ) x x x m m m m. (x x x )(x ). (y y )(y ) x x 0x y x y x 9. x x 0. x x x x x x x. (r r r ) (r ). (t t t ) (t ) r r t t t p. p p h. h h h p h p p p h h. GEOMETRY The area of a rectangle is x x square feet. The length of the rectangle is x feet. What is the width of the rectangle? x ft m. GEOMETRY The area of a triangle is x x x x square meters. The length of the base of the triangle is x meters. What is the height of the triangle? x x m Glencoe/McGraw-Hill Glencoe Algebra

24 - Reading to Learn Mathematics Dividing Polynomials Pre-Activity How can you use division of polynomials in manufacturing? Reading the Lesson Read the introduction to Lesson - at the top of page in your textbook. Using the division symbol (), write the division problem that you would use to answer the question asked in the introduction. (Do not actually divide.) (x x) (x). a. Explain 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 example that uses the division algorithm for polynomials. x x x x x x x x Which of the following is the correct way to write the quotient? C A. x B. x C. x D. x x. If you use synthetic division to divide x x x by x, the division will look like this: 0 0 Which of the following is the answer for this division problem? B A. x x B. x x x C. x x x D. x x x x Lesson - Helping You Remember. 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 exponents 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. Glencoe/McGraw-Hill Glencoe Algebra

25 - Enrichment Oblique Asymptotes The graph of y ax b, where a 0, is called an oblique asymptote of y f(x) if the graph of f comes closer and closer to the line as x or x. is the mathematical symbol for infinity, which means endless. For f(x) x, y x is an oblique asymptote because x f(x) x x, and 0 as x or. In other words, as x x increases, the value of gets smaller and smaller approaching 0. x Example Find the oblique asymptote for f(x) x x. x Use synthetic division. y x x x x x As x increases, the value of x gets smaller. In other words, since 0 as x or x, y x is an oblique asymptote. x Use synthetic division to find the oblique asymptote for each function.. y x x x. y x x x. y x x x. y ax bx c x d. y ax bx c x d Glencoe/McGraw-Hill Glencoe Algebra

26 - Study Guide and Intervention Factoring Polynomials Factor Polynomials For any number of terms, check for: greatest common factor Techniques for Factoring Polynomials 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 acx (ad bc)x bd (ax b)(cx d) For four terms, check for: Grouping ax bx ay by x(a b) y(a b) (a b)(x y) Example Factor x x. First factor out the GCF to get x x (x x ). To find the coefficients of the x terms, you must find two numbers whose product is () 0 and whose sum is. The two coefficients must be 0 and. Rewrite the expression using 0x and x and factor by grouping. x x x 0x x Group to find a GCF. x(x ) (x ) Factor the GCF of each binomial. (x )(x ) Distributive Property Thus, x x (x )(x ). Exercises Factor completely. If the polynomial is not factorable, write prime.. x y xy. mn m n. x x xy (x y) (m )(n ) (x )(x ) Lesson -. x. x y 0x y. r 0 (x )(x )(x ) x y(y x) (r )(r r ). 00m 9. x x 9. c c c c (0m )(0m ) prime c(c ) (c ) Glencoe/McGraw-Hill Glencoe Algebra

27 - Simplify Quotients In the last lesson you learned how to simplify the quotient of two polynomials by using long division or synthetic division. Some quotients can be simplified by using factoring. Example x x x x Study Guide and Intervention (continued) Factoring Polynomials x Simplify x. x x (x )( x ) (x )(x ) Factor the numerator and denominator. x Divide. Assume x,. x Exercises Simplify. Assume that no denominator is equal to 0. x. x x. x. x x x x x x x x x x 0 x x x x x. x x. x. x x 0 x x x x x x x 9x x x x x x. x x. x 9. x x x x x x x x x x 0 x x x x x 0. x x. x. x x x x 9 x x x x x x 9 x x x x x. x. x. x x x x 9 x x x x x 9 x x x x x. x y.. x y y 0 x y y x x y x 9x 9x x x Glencoe/McGraw-Hill Glencoe Algebra

28 - Skills Practice Factoring Polynomials Factor completely. If the polynomial is not factorable, write prime.. x x. 9x x x(x ) 9x (x ). x x y xy. j k jk x(x xy y ) prime. a a. ak a k (a 9)(a ) (a )(k ). b b. z z 0 (b )(b ) prime 9. m m 0. x x (m )(m 9) (x )(x ). z z. p p (z )(z ) (p )(p ). y y. c 00 (y )(y ) (c 0)(c 0). f. d d (f )(f ) (d ). 9x. y y prime (y 9) 9. n 0. m (n )(n n ) (m )(m )(m ) Lesson - Simplify. Assume that no denominator is equal to 0. x x. x x. x x x x x x 9 x x. 0x x. x x x x x 9 x x x x Glencoe/McGraw-Hill 9 Glencoe Algebra

29 - Practice (Average) Factoring Polynomials Factor completely. If the polynomial is not factorable, write prime.. a b 0ab. st 9s t s t. x y x y xy ab(a b) st(t s st) xy(x y x ). x y x y xy xy. t r rt. x xy x y xy(x x y y ) ( r)( t) (x )(x y). y 0y 9. ab a b 9. n n (y )(y ) (a )(b ) (n )(n ) 0. x x. x x. p p (x )(x ) prime (p 9)(p ). r r r. a a. c 9 r(r 9)(r ) (a )(a ) (c )(c ). x. r 9. b (x )(x x ) (r )(r ) (b 9)(b )(b ) 9. m prime 0. t t t t(t ). y y y (y )(y y 9). x (9x )(x )(x ) Simplify. Assume that no denominator is equal to 0. x. x x x. x x 9. x x 0 x x x x x (x ) x x 9. 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 expression that represents the area of the cross section of the brace. x(0. x) cm cm x cm. cm x 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 expression for the area of the cross section of the flywheel outside the crankshaft. (r r )(r r ) r r Glencoe/McGraw-Hill 0 Glencoe Algebra

30 - Reading to Learn Mathematics Factoring Polynomials Pre-Activity How does factoring apply to geometry? Read the introduction to Lesson - at the top of page 9 in your textbook. 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 x y cannot be factored, it is an example of a polynomial. prime. On an algebra quiz, Marlene needed to factor x x 0. She wrote the following answer: (x )(x ). 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 (x )(x ) x x 0. a. If you were Marlene s teacher, how would you explain 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 x can be factored further as (x ). 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 expression that is being factored. In the trinomial factor, the operation sign before the middle term is the opposite of the sign in the expression that is being factored. The sign before the last term is always a plus. Lesson - Glencoe/McGraw-Hill Glencoe Algebra

31 - Enrichment 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 extended to other odd powers. Study these examples. Example Factor a b. Extend 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 Example Factor a b. Extend 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 a b a b ab b a b In general, if n is an odd integer, when you factor a n b n or a n b n, one factor will be either (a b) or (a b), depending on the sign of the original expression. The other factor will have the following properties: The first term will be a n and the last term will be b n. The exponents of a will decrease by as you go from left to right. The exponents of b will increase by as you go from left to right. The degree of each term will be n. If the original expression was a n b n, the terms will alternately have and signs. If the original expression was a n b n, the terms will all have signs. Use the patterns above to factor each expression.. a b. c 9 d 9. e f To factor x 0 y 0,change it to (x y )(x y ) and factor each binomial. Use this approach to factor each expression.. x 0 y 0. a b Glencoe/McGraw-Hill Glencoe Algebra

32 - Study Guide and Intervention 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. nth Root Real nth Roots of b, n b, b n For any real numbers a and b, and any positive integer n, if a 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.. If n is odd and b 0, then b has one positive root.. 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. Example Example Simplify 9z. 9z (z ) z z must be positive, so there is no need to take the absolute value. Exercises Simplify. Simplify (a ) (a ) [(a ) ] (a )... p 9 p. a 0. p 0. m n 9 a p m n. b. a b 0 9. x b a b x 0. (k). 9r. p k r p. y z. q. 00x y z. y z q 0 x y z p 0 0. not a real number 0. p 9. (x) 0. (y ). (a b) x y a b. (x ). (m ). x x x (m ) x Lesson - Glencoe/McGraw-Hill Glencoe Algebra

33 - Study Guide and Intervention (continued) Roots of Real Numbers Approximate Radicals with a Calculator Irrational Number a number that cannot be expressed as a terminating or a repeating decimal Radicals such as and are examples of irrational numbers. Decimal approximations for irrational numbers are often used in applications. These approximations can be easily found with a calculator. Example.. Approximate. with a calculator. Exercises Use a calculator to approximate each value to three decimal places , , LAW ENFORCEMENT The formula r L is used by police to estimate the speed r in miles per hour of a car if the length L of the car s skid mark is measures in feet. Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skid mark 00 feet long.. mi/h 0. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h miles above Earth can be approximated by d 000h. h What is the distance to the horizon if a satellite is orbiting 0 miles above Earth? about 00 ft Glencoe/McGraw-Hill Glencoe Algebra

34 - Skills Practice Roots of Real Numbers Use a calculator to approximate each value to three decimal places Simplify (). not a real number y y. s s. x x. a a. m n m n. 00p q 0p q. w v w v. (c) 9c. (a ) b a b Lesson - Glencoe/McGraw-Hill Glencoe Algebra

35 - Use a calculator to approximate each value to three decimal places (0.9) Simplify Practice (Average) Roots of Real Numbers x 0 9. (a) x a. 9m t m.. r w. (x) m t m r w x 0. (a ) not a real number. s. p q 9. x y. x y 9 s pq x y x y 9. m n 0. x y 0. (m ). (x ) m n xy m x. 9a b 0. (x ).. x 0x d a b (x ) d x. 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. 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, feet, and 0 feet, what is the area of the garden? Round your answer to the nearest whole number. ft Glencoe/McGraw-Hill Glencoe Algebra

36 - 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 textbook. Suppose the length of a wave is feet. Explain 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 index. a. radicand: index: b. x radicand: x index: c. radicand: index:. 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 x, you must take the absolute value of x 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. Lesson - Glencoe/McGraw-Hill Glencoe Algebra

37 - Enrichment Approximating Square Roots Consider the following expansion. ( 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 approximation. ( a b a ) a b b a a a b Suppose a number can be expressed as a b, a b. Then an approximate value b of the square root is a a.you should also see that a b a a. b Example Use the formula a b b a to approximate 0 and a. a b. Let a 0 and b. Let a and b. 0 0 ( 0) ( ) Use the formula to find an approximation for each square root to the nearest hundredth. Check your work with a calculator ,, b. Show that a a a b for a b. Glencoe/McGraw-Hill Glencoe Algebra

38 - Study Guide and Intervention Radical Expressions Simplify Radical Expressions 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. Lesson - Quotient Property of Radicals For any real numbers a and b 0, and any integer n, n a a n n, if all roots are defined. b 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 exact root. Example Example Simplify a. b a b () a a (b ) b ab a b x y x y (x) x (y ) y (x) x (y ) y x x y y x Simplify. y Quotient Property Factor into squares. Product Property Simplify. x x y Rationalize the y y y denominator. x 0xy Simplify. y Exercises Simplify... a 9 b 0 a b a. x y x y y... pq p a b a bb p q Glencoe/McGraw-Hill 9 Glencoe Algebra

39 - Study Guide and Intervention (continued) Radical Expressions Operations with Radicals When you add expressions containing radicals, you can add only like terms or like radical expressions. Two radical expressions are called like radical expressions if both the indices and the radicands are alike. To multiply radicals, use the Product and Quotient Properties. For products of the form (ab cd) (ef gh), use the FOIL method. To rationalize denominators, use conjugates. Numbers of the form ab cd and ab cd, where a, b, c, and d are rational numbers, are called conjugates. The product of conjugates is always a rational number. Example Simplify Factor using squares. Simplify square roots. Multiply. Combine like radicals. Example Simplify ( )( ). Example ( )( ) 0 Exercises Simplify Simplify. () () ( ). ( 0 ) 9. ( )( ). ( )( ) 9. ( )(0 ) Glencoe/McGraw-Hill 0 Glencoe Algebra

40 - Simplify. Skills Practice Radical Expressions.... Lesson -. 0x 0x x. a b ab. d f f d f.s t s t g z g0gz. ()() z. ()(0) ( )( ) 9. ( )( ) 0. ( )( ). ( )... 0 Glencoe/McGraw-Hill Glencoe Algebra

41 - Simplify. Practice (Average) Radical Expressions t w t w. v z v z z 9. g k gk k 0. x y xy x c d c d 9a a d a.. a b 9a b a. ()(). ()() ( ) 0. ( ). ( )( ). ( 0)( 0) 0 0. ( )( ). ( ). (0 ) x x x x x. 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 expression for s if. Then evaluate s to the nearest mile per hour. 0 ; mi/h. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can be represented by the expressions x y and 9x y. Use the Pythagorean Theorem to find a simplified expression for the measure of the hypotenuse. x y Glencoe/McGraw-Hill Glencoe Algebra

42 - Reading to Learn Mathematics Radical Expressions Pre-Activity How do radical expressions apply to falling objects? Read the introduction to Lesson - at the top of page 0 in your textbook. Describe how you could use the formula given in your textbook 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. Lesson - Reading the Lesson. Complete the conditions that must be met for a radical expression to be in simplified form. The index 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 expressions used for? to rationalize binomial denominators b. How would you use a conjugate to simplify the radical expression? Multiply numerator and denominator by. c. In order to simplify the radical expression 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 explain it to another person. When rationalizing the denominator in the expression, many students think they should multiply numerator and denominator by. How would you explain 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, which equals. Glencoe/McGraw-Hill Glencoe Algebra

43 - Enrichment Special Products with Radicals Notice that ()(), or (). In general, (x) x when x 0. Also, notice that (9)(). In general, (x)(y) xy when x 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 Example Find the product: ( )( ). ( )( ) () () Example Evaluate ( ). ( ) () () Multiply. (). ( )( ). (0 )(0 ). (x )(x ). ( ). (000 0). (y )(y ). (0 x). (x 0) You can extend these ideas to patterns for sums and differences of cubes. Study the pattern below. ( x)( x x ) x x Multiply. 9. ( )( 0 ) 0. ( y w)( y yw w ).( 0)( 0 0 ). ( )( ) Glencoe/McGraw-Hill Glencoe Algebra

44 - Study Guide and Intervention Rational Exponents Rational Exponents and Radicals Definition of b For any real number b and any positive integer n, n b n b, n except when b 0 and n is even. Definition of b m n For any nonzero real number b, and any integers m and n, with n, b m n b n m ( b) n m, except when b 0 and n is even. Example Example Notice that 0. Write in radical form. Evaluate. Notice that 0, 0, and is odd. Lesson - Exercises Write each expression in radical form Write each radical using rational exponents... a b. p a b p Evaluate each expression (0.000) (0.) Glencoe/McGraw-Hill Glencoe Algebra

45 - Study Guide and Intervention (continued) Rational Exponents Simplify Expressions All the properties of powers from Lesson - apply to rational exponents. When you simplify expressions with rational exponents, leave the exponent in rational form, and write the expression with all positive exponents. Any exponents in the denominator must be positive integers When you simplify radical expressions, you may use rational exponents to simplify, but your answer should be in radical form. Use the smallest index possible. Example Example y y y y Exercises Simplify y y. x (x ) ( x ) Simplify x. ( ) ( ) (x ) x x (x) xx Simplify each expression.. x x. y. p p 0 x y p. m. x x. s m x s 9 p.. a a 9. p p a x x x x x.. ab. ab x a b b Glencoe/McGraw-Hill Glencoe Algebra

46 - Skills Practice Rational Exponents Write each expression in radical form.... or ( ). (s ) ss Write each radical using rational exponents..... xy x y Lesson - Evaluate each expression () Simplify each expression.. c c c. m 9 m 9 m 9. q q 0. p p p. x y x. x x x y.. y y.. x n n n n n 9a b a b Glencoe/McGraw-Hill Glencoe Algebra

47 - Write each expression in radical form.... m. (n ) Practice (Average) Rational Exponents or ( ) m or ( m) n n Write each radical using rational exponents m n. a 0 b 9 m n a b Evaluate each expression () Simplify each expression.. g g g 9. s s s 0. u u. y y y b. b. q.. b q t t. 0.. q 0 t 9. t z z a b z z 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.. amps. BUSINESS A company that produces DVDs uses the formula C n 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. $9 Glencoe/McGraw-Hill Glencoe Algebra

48 - Reading to Learn Mathematics Rational Exponents Pre-Activity How do rational exponents apply to astronomy? Read the introduction to Lesson - at the top of page in your textbook. The formula in the introduction contains the exponent. 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 exponents. For any real number b and for any positive integer n, b n except 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 m, except when b 0 and n is even. n ( n b) m n b Lesson -. Complete the conditions that must be met in order for an expression with rational exponents to be simplified. It has no negative exponents. It has no fractional exponents in the denominator. It is not a complex fraction. The index of any remaining radical is the least number possible.. Margarita and Pierre were working together on their algebra homework. One exercise asked them to evaluate the expression. Margarita thought that they should raise to the fourth power first and then take the cube root of the result. Pierre thought that they should take the cube root of 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 exponent gives the power and which part gives the root. How can your knowledge of integer exponents help you to keep this straight? Sample answer: An integer exponent can be written as a rational exponent. For example,. 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. Glencoe/McGraw-Hill 9 Glencoe Algebra

49 - Enrichment Lesser-Known Geometric Formulas Many geometric formulas involve radical expressions. 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 and c. c A a c A.0. The area of an equilateral triangle with a side of length a. Find A when a. A a A.. The area of a regular pentagon with a side of length a. Find A when a. A a 0 A.. The area of a regular hexagon with a side of length a. Find A when a 9. A a A 0.. The volume of a regular tetrahedron with an edge of length a. Find V when a. a V V 0.9. The area of the curved surface of a right cone with an altitude of h and radius of base r. Find S when r and h. S rr h S.. Heron s Formula for the area of a triangle uses the semi-perimeter s, where s a b c. The sides of the triangle have lengths a, b, and c. Find A when a, b, and c. A s(s a)(s b)(s c) A. The radius of a circle inscribed in a given triangle also uses the semi-perimeter. Find r when a, b, and c 9. s(s a)(s b)(s c) r s r.9 Glencoe/McGraw-Hill 0 Glencoe Algebra

50 - Study Guide and Intervention 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 index of the radical. Solve the resulting equation. Check your solution in the original equation to make sure that you have not obtained any extraneous roots. Example Example Solve x. x Original equation x Add to each side. x Isolate the radical. x Square each side. x x Divide each side by. Check () () The solution x checks. Subtract from each side. Solve x x. x x Original equation x x x Square each side. x x Simplify. x x Isolate the radical. x x Square each side. x x 0 x(x ) 0 Factor. x 0 or x Check Subtract x from each side. (0), but (0), so 0 is not a solution. (), and (), so the solution is x. Lesson - Exercises Solve each equation.. x. x. x no solution. x. x 9 no solution. x 0. x 0. 0 x 9. xx x 9. no solution 0. x 0. x x x. 9x x, Glencoe/McGraw-Hill Glencoe Algebra

51 - Study Guide and Intervention (continued) Radical Equations and Inequalities Solve Radical Inequalities A radical inequality is an inequality that has a variable in a radicand. Use the following steps to solve radical inequalities. Step Step Step If the index 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. Example Solve 0x. Since the radicand of a square root must be greater than or equal to zero, first solve 0x 0. 0x 0 0x x Now solve 0x. 0x Original inequality 0x Isolate the radical. 0x 0x 0 x Divide each side by 0. Eliminate the radical by squaring each side. Subtract from each side. It appears that x is the solution. Test some values. x x 0 x 0( ) is not a real 0(0), so the 0()., so number, so the inequality is inequality is satisfied. the inequality is not not satisfied. satisfied Therefore the solution x checks. Exercises Solve each inequality.. c. x c x x. 0x 9. x. x. x x x x 0. 9 x. 9. x x x x x 0. x. d d. b b 0 x 0 d b Glencoe/McGraw-Hill Glencoe Algebra

52 - Skills Practice Radical Equations and Inequalities Solve each equation or inequality.. x. x. j. v 0 no solution. y no solution. w. b. n 9. r 0. p. k 0. (d ) Lesson -. (t ). ( u) 0 9. z z no solution. g g. x x no solution. s s 9. x x 0. a a. r 0 r. x x 0. y y. r r Glencoe/McGraw-Hill Glencoe Algebra

53 - Practice (Average) Radical Equations and Inequalities Solve each equation or inequality.. x. x 9. p 0. h 0. c 9 9. h no solution. d. w 9. q 9 0. y 9 0 no solution. m 0. m. n. t. t. (v ) no solution. (g ). (u ) 0 9. d d 0. r r. x x 0. x x no solution. a a. z z. q no solution. a a. 9 c c. x x 9. 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 Glencoe/McGraw-Hill Glencoe Algebra

54 - Reading to Learn Mathematics Radical Equations and Inequalities Pre-Activity How do radical equations apply to manufacturing? Read the introduction to Lesson - at the top of page in your textbook. Explain how you would use the formula in your textbook 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 extraneous 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 b. Describe two ways you can check the proposed solutions of a radical equation in order to determine whether any of them are extraneous 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 extraneous. Lesson -. Complete the steps that should be followed in order to solve a radical inequality. Step If the index of the root is even, identify the values of the variable for which the radicand is nonnegative. Step Solve the Step Test inequality algebraically. values to check your solution. Helping You Remember. One way to remember something is to explain 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 explain 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 example, the only solution of x is, but the squared equation x has two solutions, and. Glencoe/McGraw-Hill Glencoe Algebra

55 - Enrichment 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 complex statement is true. Example Under what conditions is P Q true? Create the truth table for the statement. Use the information from the truth table above for P Q to complete the last column. P Q P P Q T T F T T F F F F T T T F F T T The truth table indicates that P Q is true in all cases except where P is true and Q is false. Use truth tables to determine the conditions under which each statement is true.. P Q. P (P Q). (P Q) (P Q). (P Q) (Q P). (P Q) (Q P). (P Q) (P Q) Glencoe/McGraw-Hill Glencoe Algebra

56 -9 Study Guide and Intervention Complex Numbers Add and Subtract Complex Numbers Complex Number A complex 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 (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 Complex Numbers (a bi) (c di) (a c) (b d )i Example Example ( i) ( i) ( ) ( )i 0 i Simplify ( i) ( i). ( i) ( i) ( ) [ ()]i i Simplify ( i) ( i). To solve a quadratic equation that does not have real solutions, you can use the fact that i to find complex solutions. Example Solve x 0. x 0 Original equation x x Divide each side by. x x i Exercises Simplify. Subtract from each side. Take the square root of each side.. ( i) ( i). ( i) ( i). ( i) ( i) i i 0 i Lesson -9. ( i) ( i). ( i) ( i). ( i) ( i) 9i i. ( i) ( i). (9 i) ( i) 9. ( i) ( i) i i i 0. i. i. i i Solve each equation.. x 0. x 0. 9x 9 i i i Glencoe/McGraw-Hill Glencoe Algebra

57 -9 Study Guide and Intervention (continued) Complex Numbers Multiply and Divide Complex Numbers Multiplication of Complex Numbers Use the definition of i and the FOIL method: (a bi)(c di) (ac bd ) (ad bc)i To divide by a complex number, first multiply the dividend and divisor by the complex conjugate of the divisor. Complex Conjugate a bi and a bi are complex conjugates. The product of complex conjugates is always a real number. Example Simplify ( i) ( i). ( i) ( i) () (i) (i)() (i)(i) i 0i 0i i 0() i FOIL Multiply. Simplify. Standard form Example i Simplify. i i i i i Use the complex conjugate of the divisor. i i 9i i i 9i i Multiply. i i Standard form Exercises Simplify.. ( i)( i) i. ( i)( i) i. ( i)( i) 0i. ( i)( i). ( i)( i) i. ( i)( i) i. ( i)( i)( i). ( i)( i)( i) 9. ( i)( i)( i) i i i i i 0. i. i. i i i i i i i. i. i. i i i i i i i. i. i. i i i i Glencoe/McGraw-Hill Glencoe Algebra

58 -9 Simplify. Skills Practice Complex Numbers. i. 9 i. x 9 x i.. (i)(i)(i) 0i. i i. i i. ( i) ( i) i 9. ( i) ( i) i 0. (0 i) ( i) i. ( i)( i) i. ( i)( i) i. ( i)( i) i. ( i)( i) i i i.. i i Solve each equation. i 0. x 0 i. x 0 i Lesson x 0 0 i 0. x 0 i. x 0 i. x 9 0 i Find the values of m and n that make each equation true.. 0 i m ni,. m i ni,. ( m) ni 9 i,. ( n) (m )i i, Glencoe/McGraw-Hill 9 Glencoe Algebra

59 -9 Simplify. Practice (Average) Complex Numbers. 9 i. i. s s i. a b.. a b ia. (i)(i)(i). (i) (i) 9. i 0i 9i 0. i. i 9. ( i) ( i) i i 0i. ( i) (9 i). ( i) ( i). (0 i) ( 0i) i 9i i. ( i) (0 0i). ( i)( i). ( i)( i) i i 9. ( i)( i) 0. ( i)(9 i). i i i i i. i.. i i i i i i i Solve each equation.. n 0 i. m 0 0 i. m 0 i9. m 0 i 9. m 0 i 0. x 0 i Find the values of m and n that make each equation true.. i m ni,. ( m) ni i, 9. (m ) ( n)i i,. ( n) (m 0)i i,. ELECTRICITY The impedance in one part of a series circuit is j ohms and the impedance in another part of the circuit is j ohms. Add these complex numbers to find the total impedance in the circuit. j ohms. 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 Glencoe/McGraw-Hill 90 Glencoe Algebra

60 -9 Reading to Learn Mathematics Complex Numbers Pre-Activity How do complex numbers apply to polynomial equations? Reading the Lesson. Complete each statement. Read the introduction to Lesson -9 at the top of page 0 in your textbook. Suppose the number i is defined such that i. Complete each equation. i (i) i a. The form a bi is called the standard form of a complex number. b. In the complex number i, the real part is and the imaginary part is. This is an example of a complex number that is also a(n) number. c. In the complex number, the real part is and the imaginary part is 0. This is example of complex number that is also a(n) number. d. In the complex number i, the real part is 0 and the imaginary part is. This is an example of a complex number that is also a(n) imaginary real pure imaginary number.. Give the complex conjugate of each number. a. i b. i i i. Why are complex conjugates used in dividing complex numbers? The product of complex conjugates is always a real number. Lesson -9. Explain how you would use complex conjugates to find ( i) ( 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 expression 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. Glencoe/McGraw-Hill 9 Glencoe Algebra

61 -9 Enrichment Conjugates and Absolute Value When studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z x yi.we denote the conjugate of z by z. Thus, z x yi. We can define the absolute value of a complex number as follows. z x yi x y There are many important relationships involving conjugates and absolute values of complex numbers. Example Let z x yi. Then, z (x yi)(x yi) x y (x ) y z Show z zz for any complex number z. Example Show complex number z. z z We know z zz. If z 0, then we have z z Thus, z is the multiplicative inverse of z. z is the multiplicative inverse for any nonzero z. For each of the following complex numbers, find the absolute value and multiplicative inverse.. i. i. i. i. i. i. i. i 9. i Glencoe/McGraw-Hill 9 Glencoe Algebra

62 NAME DATE PERIOD Chapter Test, Form SCORE Write the letter for the correct answer in the blank at the right of each question.. Simplify (x 0 ) (x ). A. x B. x C. x D. x. y. Simplify. yz Assume that no variable equals 0. z A. y B. y z C. y z D. y z.. Scientists have determined that the speed at which light travels is approximately 00,000,000 meters per second. Express this speed in scientific notation. A. 0 m/s B. 0 m/s C. 0 m/s D. 0 0 m/s.. Simplify (m 9) (m ). A. 9m B. m C. 9m D. 0m.. Simplify x(x y). A. x xy B. x y C. x y D. x xy.. Simplify (x x ) (x ). A. x x 0 B. x C. x D. x x x.. Which represents the correct synthetic division of (x x ) (x )? A. B. 9 C. D.. 9. Factor m 9m completely. A. m(m ) B. (m )(m ) C. (m )(m ) D. m(m 9). Assessment t 9. Simplify t t. Assume that the denominator is not equal to 0. t 0 A. t B. t C. t D. t 9. t t t t Glencoe/McGraw-Hill 9 Glencoe Algebra

63 NAME DATE PERIOD Chapter Test, Form (continued) 0. Simplify. A. B. C. D. 0.. Use a calculator to approximate to three decimal places. A..0 B..9 C..9 D..9.. Simplify. A. B. C. D... Simplify ( )( ). A. B. C. D... Simplify. A. B. C. 0 D... Write the expression in radical form. A. B. C. D... Simplify the expression m m. A. m B. m C. m D. m.. Solve x. A. B. C. D... Solve x. A. x B. x C. x D. x. 9. Simplify ( i)( i). A. i B. C. i D. i ELECTRICITY The total impedance of a series circuit is the sum of the impedances of all parts of the circuit. A technician determined that the impedance of the first part of a particular circuit was j ohms. The impedance of the remaining part of the circuit was j ohms. What was the total impedance of the circuit? A. j ohms B. j ohms C. j ohms D. j ohms 0. Bonus Find the value of k so that x divides x x 9x k with no remainder. B: Glencoe/McGraw-Hill 9 Glencoe Algebra

64 NAME DATE PERIOD Chapter Test, Form A SCORE Write the letter for the correct answer in the blank at the right of each question.. Simplify (a 0 b )(a b ). A. b a B. b a C. b D.. a a b c. Simplify a. bc Assume that no variable equals 0. A. a b c B. a b c C. a c b D. a b c.. Neptune is approximately. 0 9 kilometers from the Sun. If light travels at approximately.0 0 kilometers per second, how long does it take light from the Sun to reach Neptune? Express your answer in scientific notation. A. 0 s B.. 0 s C.. 0 s D. 00 s.. Simplify (a a a) (a a ). A. a a a B. a a a C. a a a D. a a a.. Simplify (m ). A. 9m B. 9m C. 9m m D. 9m 0m.. Simplify (x x x ) (x ). A. x x B. x x 9 x C. x D. x x x. x b. Which represents the correct synthetic division of (x x 0) (x )? A. 0 B. 0 C. 0 0 D Assessment. Factor y completely. A. (y ) B. (y )(y y ) C. (y )(y ) D. (y )(y y ). Glencoe/McGraw-Hill 9 Glencoe Algebra

65 NAME DATE PERIOD Chapter Test, Form A (continued) 9. Simplify x x A. x x 0. Simplify n w. x. Assume that the denominator is not equal to 0. 9x B. x x C. x x D. x 9. x A. n w B. n w C. n w D. n w 0.. Use a calculator to approximate to three decimal places. A.. B..00 C..0 D.... Simplify x. A. x B. x C. x x D. x x.. Simplify 0. A. B. C. 0 D... Simplify. A. B. C. D.. Write the radical y using rational exponents.. A. y B. y C. y D. y.. Simplify the expression. A. m B. m C. m D. m.. A correct step in the solution of the equation (m ) is. A. (m ) B. m C. (m ) D. m.. Solve x. A. x 0 B. x C. x D. x. 9. Simplify ( i) ( i). A. B. C. i D. i Simplify i. i m m A. i B. i C i D. 9 i 0. 9 Bonus Find the value of k so that (x x kx ) (x ) has remainder. B: Glencoe/McGraw-Hill 9 Glencoe Algebra

66 NAME DATE PERIOD Chapter Test, Form B SCORE Write the letter for the correct answer in the blank at the right of each question.. Simplify (x 0 y )(x y). A. x y B. y9 C. x D. x. y x. Simplify x z xy. yz Assume that no variable equals 0. y A. x B. z x y z C. x y z y D. z x.. Saturn is approximately. 0 9 kilometers from the Sun. If light travels at approximately.0 0 kilometers per second, how long does it take light from the Sun to reach Saturn? Express your answer in scientific notation. A.. 0 s B. 00 s C.. 0 s D. 0 s.. Simplify (x x ) (x x ). A. x x x B. x x C. x x D. x x x.. Simplify (x ). A. x B. x 0x C. x 0x D. x x.. Simplify (x x x ) (x ). A. x 9 x B. x x x x C. x x D. x x 9 x. x. Which represents the correct synthetic division of (x x ) (x )? A. B. C. 0 D Assessment. Factor x completely. A. (x )(9x x ) B. (x )(9x x ) C. (x ) D. (x )(9x x ). Glencoe/McGraw-Hill 9 Glencoe Algebra

67 NAME DATE PERIOD Chapter Test, Form B (continued) 9. Simplify x x A. x x 0. Simplify p q. x. Assume that the denominator is not equal to 0. x B. x x C. x x D. x 9. x A. p q B. p q C. p q D. p q 0.. Use a calculator to approximate 0 to three decimal places. A.. B..9 C.. D..9.. Simplify t. A. t t B. t t C. t t D. tt.. Simplify 0. A. B. C. D... Simplify. A. 0 B. 0 C. 0 D. 0.. Write the radical m using rational exponents. A. m B. m C. m D. m.. Simplify the expression. t A. t B. t 0 C. t 9 0 t D. t 0.. A correct step in the solution of the equation (z ) is. A. z B. (z ) C. (z ) 9 D. z.. Solve x. A. x 0 B. x 0 C. x 0 D. x. 9. Simplify ( i) ( i). A. 0i B. i C. 0i D Simplify i. i A. i B. i C. i D. i 0. Bonus Factor x z y y z 9x completely. B: Glencoe/McGraw-Hill 9 Glencoe Algebra

68 NAME DATE PERIOD Chapter Test, Form C SCORE Simplify. Assume that no variable equals 0.. (r t) (r 0 t ). a bc. a bc. For Questions, simplify.. (c c ) (c c ).. (9p p) (p p ).. (x )(x ) x y.. a b ( )( ) 0.. ( i) ( i).. ( i)( i).. Evaluate ( 0 )(. 0 9 ). Express the result in. scientific notation.. Use long division to find (0y 9y y 0) (y ).. Assessment. Use synthetic division to find (x x x 0) (x )... Factor xz yz x y completely. If the polynomial. is not factorable, write prime. Glencoe/McGraw-Hill 99 Glencoe Algebra

69 NAME DATE PERIOD Chapter Test, Form C (continued) x. Simplify x. Assume that the denominator. x is not equal to 0.. TREES The diameter of a tree d (in inches) is related to its. basal area BA (in square feet) by the formula d (B A). If the basal area of a tree is. square feet, what is the diameter of the tree? Use a calculator to approximate your answer to three decimal places. 9. Write the radical m using rational exponents. 9. x 0. Simplify the expression. x x 0.. Solve m... Solve y 0... POPULATION In 000, the population of New York City. was approximately,000,000. Its total area is about 00 square miles. What was the population density (number of people per square mile) of New York City in 000? Express your answer in scientific notation.. ELECTRICITY The total impedance of a series circuit is. the sum of the impedances of all parts of the circuit. Suppose that the first part of a circuit has an impedance of j ohms and that the total impedance of the circuit is j ohms. What is the impedance of the remainder of the circuit?. ELECTRICITY In an AC circuit, the voltage E (in volts),. current I (in amps), and impedance Z (in ohms) are related by the formula E I Z. Find the current in a circuit with voltage 0 j volts and impedance j ohms. i i Bonus Simplify. B: i i Glencoe/McGraw-Hill 00 Glencoe Algebra

70 NAME DATE PERIOD Chapter Test, Form D SCORE Simplify. Assume that no variable equals 0.. (c d 0 ) (c d ). a b c. a bc. For Questions, simplify.. (f f 9) (f f ).. (g g ) (g g ).. (m )(m ) x y.. a b ( )( ) 0.. ( i) ( i).. ( i)( i). Evaluate( 0 )(. 0 ). Express the result in. scientific notation.. Use long division to find (x 0x 9x 0) (x ).. Assessment. Use synthetic division to find (x x 9x 0) (x )... Factor 0x x xy y completely. If the polynomial. is not factorable, write prime. Glencoe/McGraw-Hill 0 Glencoe Algebra

71 NAME DATE PERIOD Chapter Test, Form D (continued). Simplify x x 0 x. Assume that the denominator is. not equal to 0.. TREES The diameter of a tree d (in inches) is related to its. basal area BA (in square feet) by the formula d (B A). If the basal area of a tree is.9 square feet, what is the diameter of the tree? Use a calculator to approximate your answer to three decimal places. 9. Write the radical x using rational exponents. 9. x 0. Simplify the expression. 0. x x. Solve 0s... Solve t... POPULATION In 000, the population of Hong Kong. was approximately. million. Its total area is about 000 square kilometers. What was the population density (number of people per square kilometer) of Hong Kong in 000? Express your answer in scientific notation.. ELECTRICITY The total impedance of a series circuit is. the sum of the impedances of all parts of the circuit. Suppose that the first part of a circuit has an impedance of j ohms and that the total impedance of the circuit is j ohms. What is the impedance of the remainder of the circuit?. ELECTRICITY In an AC circuit, the voltage E (in volts),. current I (in amps), and impedance Z (in ohms) are related by the formula E I Z. Find the impedance in a circuit with voltage j volts and current j amps. i i Bonus Simplify. B: i i Glencoe/McGraw-Hill 0 Glencoe Algebra

72 NAME DATE PERIOD Chapter Test, Form SCORE Simplify. Assume that no variable equals 0. ). ( a a. ). x y 0 ( xy ( xy. ) For Questions, simplify.. p r pr (pr r ).. (m n).. x 0x.. x y.. x y.. 0. x 9 9. x ( i)( i) 0.. ( i) ( i) ( i).. Evaluate Express the result in scientific notation.. Assessment. Use long division to find x x x x.. x. Use synthetic division to find x x.. x Glencoe/McGraw-Hill 0 Glencoe Algebra

73 NAME DATE PERIOD Chapter Test, Form (continued) For Questions and, factor completely. If the polynomial is not factorable, write prime.. w n.. x y. m. Simplify. Assume that the. (m m )(m ) denominator is not equal to 0.. GEOMETRY The volume V of a sphere and the length of. its radius r are the related by the formula r V. Use the formula to find radius of a sphere with volume 00 cubic meters. Approximate your answer to three decimal places. 9. Write the expression x 9 y using rational exponents Simplify the expression. 0.. Solve x 0... Solve x x... Simplify i.. i. Solve x 0.. STATISTICS During fiscal year 99, total New York state. expenditures were approximately $. billion dollars. The population of New York in 99 was approximately million. Find New York s 99 per capita (per person) expenditures. Express your answer in scientific notation. Bonus For a circuit with two impedances in parallel, the total impedance of the circuit Z t is given by the equation ZZ Z t, where Z Z Z and Z are the parallel impedances. Find the total impedance of a circuit with Z j ohms and Z j ohms. B: Glencoe/McGraw-Hill 0 Glencoe Algebra

74 NAME DATE PERIOD Chapter Open-Ended Assessment SCORE Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.. Jorge works for the A-Glide Sled Company. This company estimates its monthly profit for the sale of x sleds, in hundreds of dollars, is given by the expression x 9. Tia works for a competing sled manufacturer, SnowFun. Tia s company estimates that its monthly profit for the sale of x sleds, in hundreds of dollars, is given by the expression x. Mark has been offered a job at both companies and decides he will work for the company that has the greatest monthly profit. Before he makes his decision, however, he asks Jorge and Tia the average number of sleds sold each month by each of their companies. a. Why is the number of sleds sold important to Mark? b. Assume both companies make the same number of sleds in a certain month. Determine the number of sleds that would make Mark want to work for SnowFun, and give the profit, to the nearest dollar, earned by each company during that month. c. After talking to Jorge and Tia, Mark decided to work for A-Glide. Assume that both companies average the same number of sleds sold per month. Write and solve an inequality to determine the possible responses Mark might have heard from Jorge and Tia. What does this tell you about the number of sleds sold each month?. You are given an unlimited number of tiles with the given dimensions. length: x units length: x units length: unit width: x units width: unit width: unit area: x units area: x units area: unit The polynomial x x can be represented by the figure at the right. These tiles can be arranged to form the rectangle shown. Notice that the area of the rectangle is x x units. a. Find the length and width of the rectangle. b. Explain how to find the perimeter of the rectangle. Then find the perimeter. c. Select a value for x and substitute that value into each of the expressions above. For your value of x, state the length, width, perimeter, and area of the rectangle. Discuss any restrictions on your choice of x. d. Factor the polynomial x x. e. Compare your answers to parts a and d. f. Draw a tile model for a different polynomial. Then write your polynomial and its factors. Explain how your model relates to the factors of your polynomial. x x x x x x x x x x Assessment Glencoe/McGraw-Hill 0 Glencoe Algebra

75 NAME DATE PERIOD Chapter Vocabulary Test/Review SCORE absolute value binomial coefficient complex conjugates complex number conjugates constant degree dimensional analysis extraneous solution FOIL method imaginary unit like radical expressions like terms monomial nth root polynomial power principal root pure imaginary number radical equation radical inequality rationalizing the denominator scientific notation simplify square root synthetic division terms trinomial Underline or circle the correct word or phrase to complete each sentence.. A polynomial with two terms is called a (monomial, binomial, trinomial).. i is a(n) (pure imaginary number, imaginary unit, complex number).. (Scientific notation, Synthetic division, Absolute value) is used to write very large and very small numbers without having to write many zeros.. If you square both sides of a radical equation and obtain a solution that does not satisfy the original equation, you have found a(n) (coefficient, complex number, extraneous solution).. x 0 and x 0 are (radical equations, radical inequalities, like radical expressions).. The expressions and are (complex conjugates, conjugates, like terms).. The monomials that make up a polynomial are called (conjugates, terms, principal roots).. A monomial that contains no variables is called a (power, constant, degree). 9. The expression 0 is a (trinomial, nth root, power). 0. One of the steps that may be necessary to simplify a radical expression is (synthetic division, scientific notation, rationalizing the denominator). In your own words Define each term.. square root. pure imaginary number Glencoe/McGraw-Hill 0 Glencoe Algebra

76 NAME DATE PERIOD Chapter Quiz (Lessons through ) SCORE For Questions and, simplify. Assume that no variable equals 0... (n y )(n y ). ( xy) ( xy0).. Express in scientific notation... Evaluate (. 0 )( 0 ). Express the result in. scientific notation. Simplify.. (p q) (p q).. (x ) (x ).. (x )(x ).. Standardized Test Practice Which expression is equal to (0a a )(a )? A. a B. a a C. a D. a. a a Simplify. 9. (m m ) (m ) (a a 0a ) (a ) 0. NAME DATE PERIOD Chapter Quiz (Lessons and ) For Questions and, factor completely. If the polynomial is not factorable, write prime.. c 9.. a a. SCORE Assessment. Simplify x x x. Assume that the denominator is not. equal to 0.. Simplify w 9. y.. Use a calculator to approximate to three decimal. places. Glencoe/McGraw-Hill 0 Glencoe Algebra

77 NAME DATE PERIOD Chapter Quiz (Lessons and ) SCORE For Questions, simplify.. x. 0. ( ). m n. ( )( ).. Write the expression x in radical form.. Write the radical z using rational exponents. 9. Evaluate. 0. Simplify t t NAME DATE PERIOD Chapter Quiz (Lessons and 9) SCORE Solve each equation. y y 9. v For Questions and, solve each inequality.. x. x. Solve x Simplify ( 9i) ( i) 9. ( i)( i) 0. i i Glencoe/McGraw-Hill 0 Glencoe Algebra

78 NAME DATE PERIOD Chapter Mid-Chapter Test (Lessons through ) SCORE Part I Write the letter for the correct answer in the blank at the right of each question. Simplify.. (x y) (x y ) A. 0x 0 y B. 0x C. 0x y y D. 0x y. 0. x y z xyz A. x y z y B. xz C. 9x y z D. x y z.. (x x ) (x x ) A. x x B. x x C. x x D. x x.. (s )(s ) A. s s B. s C. s s D. s s. x. x x (Assume that the denominator is not equal to 0.) x 0 A. x B. C. x x x. x 9 D. x. x A. x B. x C. x D. x.. x y z Part II A. xyz B. xy z C. xyz D. x y z.. Use long division to find (x x x ) (x ).. 9. Use synthetic division to find (x x x 9) (x ) Use a calculator to approximate to three decimal 0. places.. Some computer chips can perform a floating point. operation in less than second. Express this value in scientific notation. Assessment. Evaluate ( 0 )(. 0 ). Express your answer in. scientific notation.. Factor ax a x 0 completely. If the polynomial is. not factorable, write prime.. Simplify (x x x ) (x ).. Glencoe/McGraw-Hill 09 Glencoe Algebra

79 NAME DATE PERIOD Chapter Cumulative Review (Chapters ). Write an algebraic expression to represent the verbal. expression the square of the sum of a number and three. (Lesson -). If f(x) x x, find f( a). (Lesson -).. Write an equation in slope-intercept form of the line. through (, ) and (, ). (Lesson -). Solve the system of equations x y. x y by using elimination. (Lesson -). STORES The floor area of a furniture storeroom is. 00 square yards. One sofa requires square yards and one dining table requires square yards of space. The room can hold a maximum of 0 pieces of furniture. Let s represent the number of sofas and t represent the number of tables. Write a system of inequalities to represent the number of pieces of furniture that can be placed in the storeroom. (Lesson -) 0. State the dimensions of matrix A if A 0. Find the product 0. (Lesson -). 0, if possible. (Lesson -).. Write a matrix equation for the system of equations. m n m n 9. (Lesson -) Evaluate.. 0 Express the result in scientific notation. 9. (Lesson -) 0. Use long division to find (x x x) (x ). (Lesson -) 0.. Simplify 9x y. (Lesson -).. Write the radical z using rational exponents. (Lesson -).. Simplify i. (Lesson -). i Glencoe/McGraw-Hill 0 Glencoe Algebra

80 NAME DATE PERIOD Standardized Test Practice (Chapters ) Part : Multiple Choice Instructions: Fill in the appropriate oval for the best answer.. In the figure, circle P represents all prime numbers, circle Q represents all numbers whose square roots are not integers, and circle R represents all multiples of. In which region does belong? A. a B. b R C. c D. d. P b a c d Q A B C D. Find the reciprocal of x. E. x x F. x x G. H. x. x E F G H. Suppose a set of data contains just two data items. If the median is w, the mean is x, and the mode is y, which of the following must be equal? A. w, x, and y B. x and y C. w and x D. w and y. A B C D. What is the value of cd in the equation cd cd? E. F. G. H.. E F G H. Hoshiko owns one-fifth of a business. She sells her share for $,000. What is the total value of the business? A. $000 B. $,000 C. $00,000 D. $0,000. A B C D. If r is an odd integer and m r, then m will always be. E. odd F. even G. positive H. negative. E F G H. % of 0 is % of. A. B. 0 C. 00 D.,000.. If m, then what is the value of m? E. F. 0 G. H.. A E B F C G D H Assessment 9. Which is the equation of a line that passes through a point with coordinates (, ) and is perpendicular to the graph of y x? A. y x B. y x C. y x D. y x 9. A B C D 0. If mn and m n, then (m n). E. F. G. 00 H. cannot be determined 0. E F G H Glencoe/McGraw-Hill Glencoe Algebra

81 NAME DATE PERIOD Standardized Test Practice (continued) Part : Grid In Instructions: Enter your answer by writing each digit of the answer in a column box and then shading in the appropriate oval that corresponds to that entry.. What is the value of a b b a if.. a b?. The volume of a cube with a surface area of in. is in.. The circle is divided into eight.. sectors of equal area. In two 0 / / consecutive spins, what is the.... probability of spinning a and then spinning an 0 odd-numbered region?. For all positive integers m, m m. What is the value of x if x,00? / / / /... / / Part : Quantitative Comparison Instructions: Compare the quantities in columns A and B. Shade in A if the quantity in column A is greater; B if the quantity in column B is greater; C if the quantities are equal; or D if the relationship cannot be determined from the information given. Column A Column B.. y A B C D y (y 0) y 0. x 0. A B C D x 0 0% of x. x k. A B C D (x k) (x k). x. A B C D x Glencoe/McGraw-Hill Glencoe Algebra

82 Standardized Test Practice Student Record Sheet (Use with pages of the Student Edition.) Glencoe/McGraw-Hill A Glencoe Algebra NAME DATE PERIOD Answers Select the best answer from the choices given and fill in the corresponding oval. 9 Solve the problem and write your answer in the blank. Also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 0 Select the best answer from the choices given and fill in the corresponding oval. 0 9 D C B A D C B A D C B A D C B A D C B A / / / / / / / / / / / / / / / / D C B A D C B A D C B A D C B A D C B A D C B A D C B A D C B A D C B A Part Short Response/Grid In Part Short Response/Grid In Part Multiple Choice Part Multiple Choice Part Quantitative Comparison Part Quantitative Comparison

83 Answers (Lesson -) Lesson - - Study Guide and Intervention 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 Exponent a n a n and a n for any real number a 0 and any integer n. a n When you simplify an expression, you rewrite it without parentheses or negative exponents. The following properties are useful when simplifying expressions. 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 Properties of Powers For a, b real numbers and m, n integers: (a m ) n a mn (ab) m a m b m a n a n b, b 0 b n a n b n b or n b a, a 0, b 0 a n Example Simplify. Assume that no variable equals 0. a. (m n )(mn) (m n )(mn) m n m n m m n n m n m b. (m ) (m ) (m ) (m ) m m m m m Exercises Simplify. Assume that no variable equals 0.. c c c c. b b b. (a ) a 0 x y x y y x a b a b... b a x y xy x y. (a b ) (abc) a b c. m m m 9. m n mn m n 0. c t. j(j k )(j k j mn ). (m n) m c t m n k Glencoe/McGraw-Hill 9 Glencoe Algebra - Study Guide and Intervention (continued) Monomials Scientific Notation Scientific notation A number expressed in the form a 0 n, where a 0 and n is an integer Example Express,000,000 in scientific notation.,000,000. 0,000, Write 0,000,000 as a power of ten. Example. 0 Evaluate 0. Express the result in scientific notation Exercises Express each number in scientific notation.., ,0, ,000, , , Evaluate. Express the result in scientific notation. 0. (. 0 )( 0 ). (. 0 )( 0 ). (. 0 )( 0 ) (. 0 ). (. 0 ). (. 0 ) ASTRONOMY Pluto is,. million miles from the sun. Write this number in scientific notation. Source: New York Times Almanac. 0 9 miles 0. CHEMISTRY The boiling point of the metal tungsten is 0,0 F. Write this temperature in scientific notation. Source: New York Times Almanac.0 0. BIOLOGY The human body contains 0.000% iodine by weight. How many pounds of iodine are there in a 0-pound teenager? Express your answer in scientific notation. Source: Universal Almanac. 0 lb Glencoe/McGraw-Hill 0 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

84 Answers (Lesson -) Lesson - - Skills Practice Monomials Simplify. Assume that no variable equals 0.. b b b. c c c c 9. a a a. x x x x. (g ) g. (u) u. (x) x. (z) 0z 9. (d) d 0. (t ) t. (r ) r. s s s. k 9 k k 0. (f g) f 9 g. (x) (y) x y. gh( g h ) g h. 0x y (0xy ) 00x y. wz w z z w a bc a b c c a b pq r p q r q p Express each number in scientific notation.., ,00, Evaluate. Express the result in scientific notation ( 0 )(. 0 ) Glencoe/McGraw-Hill Glencoe Algebra Answers - Practice (Average) Monomials Simplify. Assume that no variable equals 0.. n n n. y y y y. t 9 t t. x x x. (f ) f. (b c ). (d t v )(dt v 0d ) t. u(z) uz v m 9. y m s y 0. x 9my sx x (x ) x... (w z )(w). (m n ) (m n p ) m n p 0 wz. d f d f d f 9. (x y )(xy ) (x ) y x x y r s z x y x y 0(m v)(v) (v) (m ) 9r s z.. s x y x x c 9 b v m Express each number in scientific notation. 9. 9, Evaluate. Express the result in scientific notation.. (. 0 )(.9 0 ). (. 0 9 )( ) COMPUTING The term bit, short for binary digit, was first used in 9 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. 0 miles per second, at what velocity does it travel through water? Write your answer in scientific notation..9 0 mi/s. 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 0 meters and a blue wavelength measures about. 0 meters, about how many times longer is the infrared wavelength than the blue wavelength? about. times Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

85 Answers (Lesson -) Lesson - - 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 textbook. Your textbook gives the U.S. public debt as an example from economics that involves large numbers that are difficult to work with when written in standard notation. Give an example 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 expression is a monomial or not a monomial. If it is a monomial, tell whether it is a constant or not a constant. a. x monomial; not a constant b. y y not a monomial c. monomial; constant d. z not a monomial. Complete the following definitions of a negative exponent and a zero exponent. 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 exponents that you would use to simplify each expression. (Do not actually simplify.) a. m m quotient of powers b. y 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 exponents and when to use negative ones. What is an easy way to remember this? Sample answer: Use a positive exponent if the number is 0 or greater. Use a negative number if the number is less than. Glencoe/McGraw-Hill Glencoe Algebra - Enrichment Properties of Exponents The rules about powers and exponents are usually given with letters such as m, n, and k to represent exponents. For example, one rule states that a m a n a m n. In practice, such exponents are handled as algebraic expressions and the rules of algebra apply. Example 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 exponent n as n. It is important always to collect like terms only. Example Simplify (a n b m ). (a n b m ) (a n b m )(a n b m ) F O I L a n a n a n b m a n b m b m b m The second and third terms are like terms. a n a n b m b m Simplify each expression by performing the indicated operations.. m m. (a ) n a n. ( n b ) k kn b k. (x a j ) m x m a jm. (ay n ) a y n. (b k x) b k x. (c ) hk c hk. (d n ) d n 9. (a b)(a n b ) a n b 0. (x n y m )(x m y n a ). n. a x x n x n m y n m a n x n. (ab a b)(a n b n ) a n b ab n a n b a b n. ab (a b n ab n b n ) a b n a b n ab n Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

86 Answers (Lesson -) Lesson - - Study Guide and Intervention Polynomials Add and Subtract Polynomials Polynomial a monomial or a sum of monomials Like Terms 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. Example Simplify rs r s r rs s. rs r s r rs s (r r ) (rs rs) (s s ) Group like terms. r rs s Combine like terms. Example Simplify xy xy x y (0xy xy x y). xy xy x y (0xy xy x y) xy xy x y 0xy xy x y Distribute the minus sign. (x y x y ) (xy xy ) (xy 0xy) Group like terms. x y xy xy Combine like terms. Exercises Simplify.. (x x ) (x x ). (y xy x ) (xy y x ) x x y xy x. (m m) (m m ). x y y x m m x y. (p pq q ) (p pq q ). j k j j k p pq 0q j k. (m n ) (n m ). bc b c b c bc 0m n b bc c 9. r s rs r s rs r s 9rs 0. 9xy x y (y xy x ) r s rs x xy 0y. (xy x y) (x y xy). 0.b.b. (.9b.b.) x xy y.9b.b 0.. (bc 9b c ) (c b bc). x y xy y xy 0x 0b bc c x xy y. x xy y xy y x. p p p p p p x xy y 9p p p Glencoe/McGraw-Hill Glencoe Algebra Answers - Study Guide and Intervention (continued) Polynomials Multiply Polynomials You use the distributive property when you multiply polynomials. When multiplying binomials, the FOIL pattern is helpful. To multiply two binomials, add the products of F the first terms, FOIL Pattern O the outer terms, I the inner terms, and L the last terms. Example Find y( y y ). y( y y ) y() y(y) y(y ) Distributive Property y y 0y Multiply the monomials. Example Find (x )(x ). (x )(x ) x x x () x () First terms Outer terms Inner terms Last terms x x 0x Multiply monomials. x x Add like terms. Exercises Find each product.. x(x ). a( a a ). y ( y y ) x 0x a a a y 0y y. (x )(x ). ( x)( x). (x )(x ) x x x x x x. (x )(x ). (x )(x ) 9. (x )(x 0) x x x x x x 0 0. (a c) (a c). (a )(a ). x(x ) x ( x) a c a 0a x x 0x. (t )(t ). (r ). (c )(c ) t t 0 r r 9 c c. (a )(a ). (x )(x x) a 9 x x x x. (x )(x ) 9. (x )(x x ) x x 0 x x x 0. (n )(n n ). (x )(x x ) n 0n n n x x x Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

87 Answers (Lesson -) Lesson - - Skills Practice Polynomials Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.. x b x yes;. c no. xz y d yes; Simplify.. (g ) (g ). (d ) (d ) g d. (x x ) (x x ). (f f ) (f f ) x x f f. (r r ) (r r ) 9. (x xy) (x xy y ) r 9r x xy y 0. (t ) (t t ). (u ) ( u u) t t u u 0. (c d ). x (x 9) 0c d x 9x. q(pq q ). w(hk 0h m k w ) pq q hk w 0h m w k w. m n (m n mnp ). s y(s y sy ) m n m n p m n s y 9s y s y. (c )(c ) 9. (z )(z ) c 0c z z 0. (a ). (x )(x ) a 0a x 9x. (r s)(r s). (y )(y ) r s y y. ( b)( b). (w ) 9 b 9w w Glencoe/McGraw-Hill Glencoe Algebra - Practice (Average) Polynomials Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.. x xy xy yes;. ac a d m yes;. n 9 (m n) no. x z x yes;. c c no. r s no Simplify.. (n ) (n ). (w w ) ( w ) n w w 9. (n n ) (n 9n ) 0. (x x) (x x ) 9n n x x. (m mp p ) (m mp p ). (x xy y ) (x xy y ) m mp p x xy y. (t ) (t t ). (u ) ( u u) t t u u 0. 9( y w). 9r y (ry r y r 0 ) 9y w r y 9 r y r y. a w(a w aw ). a w (a w a w 9aw ) a w a w a w 9 a w a w 9 9. x (x xy y ) 0. ab d (ab d ab) x x y x y a b d a b d. (v )(v ). (a 9y)(a y) v v a ay 9y. ( y ). (x y) y y x 0x y y. (x w)(x w). (n )(n ) x w n 9. (w s)(w ws s ). (x y)(x xy y ) w s x x y xy y 9. BANKING Terry invests $00 in two mutual funds. The first year, one fund grows.% and the other grows %. Write a polynomial to represent the amount Terry s $00 grows to in that year if x represents the amount he invested in the fund with the lesser growth rate. 0.0x GEOMETRY The area of the base of a rectangular box measures x x square units. The height of the box measures x units. Find a polynomial expression for the volume of the box. x x x units Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

88 Answers (Lesson -) Lesson - - 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 textbook. Suppose that Shenequa decides to enroll in a five-year engineering program rather than a four-year program. Using the model given in your textbook, 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 $,0 by.0. Reading the Lesson. State whether the terms in each of the following pairs are like terms or unlike terms. a. x,y unlike terms b. m,m like terms c. r,s unlike terms d., like terms. State whether each of the following expressions is a monomial, binomial, trinomial, or not a polynomial. If the expression is a polynomial, give its degree. a. r r trinomial; degree b. x not a polynomial c. x y binomial; degree d. ab ab ab 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 (x )(x ). Show the terms you would multiply, but do not actually multiply them. F (x)(x) O (x)() I ()(x) L ()() Helping You Remember. You can remember the difference between monomials, binomials, and trinomials by thinking of common English words that begin with the same prefixes. 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 Glencoe/McGraw-Hill 9 Glencoe Algebra Answers - Enrichment 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 p n p m n 0 m n p. x y z x y z x y z x y z 0. a ab b a ab b a ab b. a ab b a ab b a ab b. a ab b a b a a b ab b. a a a a a 9 a a a 9 a. x x x x 9 x 9 x 0 x 9 x 0. x x x x x x x x x x Glencoe/McGraw-Hill 0 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

89 Answers (Lesson -) Lesson - - Study Guide and Intervention 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. Example 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 p t r p qt r 9 p t r pt qr p Example x x x x x x 9 ()x x Use long division to find (x x x 9) (x ). x x ()x x x 9 ()x The quotient is x x, and the remainder is. x x x 9 x x Therefore x x. Exercises Simplify. a 0a a mn 0m n m n... 0a b b a b ab n m a 0a 0 ab a. (x x ) (x ). (m m ) (m ) x m m b a. (p ) (p ). (t t ) (t ) p p t t p t. (x ) (x ) 9. (x x x ) (x ) x x x x x x Glencoe/McGraw-Hill Glencoe Algebra - Study Guide and Intervention (continued) Dividing Polynomials Use Synthetic Division Synthetic division a procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x r Use synthetic division to find (x x x ) (x ). Step Write the terms of the dividend so that the degrees of the terms are in x x x descending order. Then write just the coefficients. Step Write the constant r of the divisor x r to the left, In this case, r. Bring down the first coefficient,, as shown. Step Multiply the first coefficient by r,. Write their product under the second coefficient. Then add the product and the second coefficient:. Step Multiply the sum,, by r:. Write the product under the next coefficient and add: (). Step Multiply the sum,, by r:. Write the product under the next coefficient and add: 0. The remainder is 0. 0 Thus, (x x x ) (x ) x x. Exercises Simplify.. (x x 9x ) (x ). (x x x ) (x ) x x x x. (x x 0x ) (x ). (x x 9x 9) (x ) x x x x x. (x 0x 9x ) (x ). (x x x ) (x ) x 9 x x 0 x x. (x 9x x ) (x ). (x x x 0) (x ) x x x x x x 9. (x x x 9) (x ) 0. (x x x x ) (x ) x x x x x x. (x 0x x 0x ) (x ) x x 0 x Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

90 Answers (Lesson -) Lesson - - Skills Practice Dividing Polynomials Simplify. 0c x 0. c. x y y y y x x x. y y. x x. (q q )(q ). (f f f f )(f ) q f f q f. (j k 9jk ) jk. (a h a h a ) (a ) j k h ah a 9. (n n 0) (n ) 0. (d d ) (d ) n d. (s s ) (s ). (y y )(y ) s y. (g 9) (g ). (x x ) (x ) g x x u u u.. u x x x u x x. (v v 0)(v ). (t t t t 0)(t ) 0 v t t v y y y y x x 9x x y y x x x. (p p p) ( p ). (c c c )(c ) p p c p c. GEOMETRY The area of a rectangle is x x x square units. The width of the rectangle is x units. What is the length of the rectangle? x x units Glencoe/McGraw-Hill Glencoe Algebra Answers - Practice (Average) Dividing Polynomials Simplify. r. 0 r 0r r r k k. m k m 9m r k r km m 9m k. (0x y x y x y) (x y). (w z w z w z) (w z) x y wz wz. (a a a )(a). (d k d k dk )(dk ) a a a d d f f 0 f x x x. f. x z w 9. (a ) (a ) a a 0. (b ) (b ) b b 9 x x x x x x. x x. x x x. (w w w ) (w ). (y y y ) (y ) w w w y y y y. (x x x x 0) (x ). (m m ) (m ) x x x m m m m. (x x x )(x ). (y y )(y ) x x 0x y x y x 9. x x 0. x x x x x x x. (r r r ) (r ). (t t t ) (t ) r r t t t p. p p h. h h h p h p p p h h m. GEOMETRY The area of a rectangle is x x square feet. The length of the rectangle is x feet. What is the width of the rectangle? x ft. GEOMETRY The area of a triangle is x x x x square meters. The length of the base of the triangle is x meters. What is the height of the triangle? x x m Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A9 Glencoe Algebra

91 Answers (Lesson -) Lesson - - 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 textbook. Using the division symbol (), write the division problem that you would use to answer the question asked in the introduction. (Do not actually divide.) (x x) (x) Reading the Lesson. a. Explain 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 example that uses the division algorithm for polynomials. x x x x x x x x Which of the following is the correct way to write the quotient? C A. x B. x C. x D. x x. If you use synthetic division to divide x x x by x, the division will look like this: 0 0 Which of the following is the answer for this division problem? B A. x x B. x x x C. x x x D. x x x x Helping You Remember. 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 exponents 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. Glencoe/McGraw-Hill Glencoe Algebra - Enrichment Oblique Asymptotes The graph of y ax b, where a 0, is called an oblique asymptote of y f(x) if the graph of f comes closer and closer to the line as x or x. is the mathematical symbol for infinity, which means endless. For f(x) x, y x is an oblique asymptote because x f(x) x x, and 0 as x or. In other words, as x x increases, the value of gets smaller and smaller approaching 0. x Example Find the oblique asymptote for f(x) x Use synthetic division. x x. y x x x x x As x increases, the value of x x gets smaller. In other words, since 0 as x or x, y x is an oblique asymptote. Use synthetic division to find the oblique asymptote for each function.. y x x x y x. y x x x y x. y x x x y x. y ax x bx c y ax b ad d. y ax x bx c y ax b ad d Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A0 Glencoe Algebra

92 Answers (Lesson -) Lesson - - Study Guide and Intervention Factoring Polynomials Factor 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 ) Techniques for Factoring Polynomials For three terms, check for: Perfect square trinomials a ab b (a b) a ab b (a b) General trinomials acx (ad bc)x bd (ax b)(cx d) For four terms, check for: Grouping ax bx ay by x(a b) y(a b) (a b)(x y) Example Factor x x. First factor out the GCF to get x x (x x ). To find the coefficients of the x terms, you must find two numbers whose product is () 0 and whose sum is. The two coefficients must be 0 and. Rewrite the expression using 0x and x and factor by grouping. x x x 0x x Group to find a GCF. x(x ) (x ) Factor the GCF of each binomial. (x )(x ) Distributive Property Thus, x x (x )(x ). Exercises Factor completely. If the polynomial is not factorable, write prime.. x y xy. mn m n. x x xy (x y) (m )(n ) (x )(x ). x. x y 0x y. r 0 (x )(x )(x ) x y(y x) (r )(r r ). 00m 9. x x 9. c c c c (0m )(0m ) prime c(c ) (c ) Glencoe/McGraw-Hill Glencoe Algebra Answers - Study Guide and Intervention (continued) Factoring Polynomials Simplify Quotients In the last lesson you learned how to simplify the quotient of two polynomials by using long division or synthetic division. Some quotients can be simplified by using factoring. Example x x x x x Simplify x. x x (x )( x ) Factor the numerator and denominator. (x )(x ) x Divide. Assume x,. x Exercises Simplify. Assume that no denominator is equal to 0. x x x x... x x x x x x x x x x 0 x x x x x x x x 0... x x x x x x x x x 9x x x x x x x x x.. 9. x x x x x x x x x x 0 x x x x x x x x 0... x x x x x x 9 x x x x 9 x x x x x x x... x 9 x x x x x x x x 9 x x x x x x x... x x x y y y 0 y y y x 9x 9x x x Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

93 Answers (Lesson -) Lesson - - Skills Practice Factoring Polynomials Factor completely. If the polynomial is not factorable, write prime.. x x. 9x x x(x ) 9x (x ). x x y xy. j k jk x(x xy y ) prime. a a. ak a k (a 9)(a ) (a )(k ). b b. z z 0 (b )(b ) prime 9. m m 0. x x (m )(m 9) (x )(x ). z z. p p (z )(z ) (p )(p ). y y. c 00 (y )(y ) (c 0)(c 0). f. d d (f )(f ) (d ). 9x. y y prime (y 9) 9. n 0. m (n )(n n ) (m )(m )(m ) Simplify. Assume that no denominator is equal to 0. x x x x x x.. x x x x 9 x x x 0x x x x x.. x x x 9 x x Glencoe/McGraw-Hill 9 Glencoe Algebra - Practice (Average) Factoring Polynomials Factor completely. If the polynomial is not factorable, write prime.. a b 0ab. st 9s t s t. x y x y xy ab(a b) st(t s st) xy(x y x ). x y x y xy xy. t r rt. x xy x y xy(x x y y ) ( r)( t) (x )(x y). y 0y 9. ab a b 9. n n (y )(y ) (a )(b ) (n )(n ) 0. x x. x x. p p (x )(x ) prime (p 9)(p ). r r r. a a. c 9 r(r 9)(r ) (a )(a ) (c )(c ). x. r 9. b (x )(x x ) (r )(r ) (b 9)(b )(b ) 9. m prime 0. t t t t(t ). y y y (y )(y y 9). x (9x )(x )(x ) Simplify. Assume that no denominator is equal to 0. x x x 0 x x x x x x x x 9... x x (x ) x x 9. 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 expression that represents the area of the cross section of the brace. x(0. x) cm x cm cm. cm x 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 expression for the area of the cross section of the flywheel outside the crankshaft. (r r )(r r ) r r Glencoe/McGraw-Hill 0 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

94 Answers (Lesson -) Lesson - - Reading to Learn Mathematics Factoring Polynomials Pre-Activity How does factoring apply to geometry? Read the introduction to Lesson - at the top of page 9 in your textbook. 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 x y cannot be factored, it is an example of a polynomial. prime. On an algebra quiz, Marlene needed to factor x x 0. She wrote the following answer: (x )(x ). 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 (x )(x ) x x 0. a. If you were Marlene s teacher, how would you explain 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 x can be factored further as (x ). 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 expression that is being factored. In the trinomial factor, the operation sign before the middle term is the opposite of the sign in the expression that is being factored. The sign before the last term is always a plus. Glencoe/McGraw-Hill Glencoe Algebra Answers - Enrichment 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 extended to other odd powers. Study these examples. Example Factor a b. Extend 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 Example Factor a b. Extend 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 a b a b ab b a b In general, if n is an odd integer, when you factor a n b n or a n b n, one factor will be either (a b) or (a b), depending on the sign of the original expression. The other factor will have the following properties: The first term will be a n and the last term will be b n. The exponents of a will decrease by as you go from left to right. The exponents of b will increase by as you go from left to right. The degree of each term will be n. If the original expression was a n b n, the terms will alternately have and signs. If the original expression was a n b n, the terms will all have signs. Use the patterns above to factor each expression.. a b (a b)(a a b a b a b a b ab b ). c 9 d 9 (c d)(c c d c d c d c d c d c d cd d ). e f (e f )(e 0 e 9 f e f e f e f e f e f e f e f ef 9 f 0 ) To factor x 0 y 0,change it to (x y )(x y ) and factor each binomial. Use this approach to factor each expression.. x 0 y 0 (x y)(x x y x y xy y )(x y)(x x y x y xy y ). a b (a b)(a a b a b a b a b ab b )(a b) (a a b a b a b a b ab b ) Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

95 Answers (Lesson -) Lesson - - Simplify Radicals Study Guide and Intervention Roots of Real Numbers Square Root For any real numbers a and b, if a b, then a is a square root of b. nth Root For any real numbers a and b, and any positive integer n, if a 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. Example Simplify 9z. Example 9z (z ) z z must be positive, so there is no need to take the absolute value. Simplify (a ) (a ) [(a ) ] (a ) Exercises Simplify.... p 9 p. a 0. p 0. m n 9 a p m n. b. a b 0 9. x b a b x 0. (k). 9r. p k r p. y z. q. 00x y z y z q 0 x y z p 0 0. not a real number 0. p 9. (x) 0. (y ). (a b) x y a b. (x ). (m ). x x x (m ) x Glencoe/McGraw-Hill Glencoe Algebra - Study Guide and Intervention (continued) Roots of Real Numbers Approximate Radicals with a Calculator Irrational Number a number that cannot be expressed as a terminating or a repeating decimal Radicals such as and are examples of irrational numbers. Decimal approximations for irrational numbers are often used in applications. These approximations can be easily found with a calculator. Example Approximate. with a calculator... Exercises Use a calculator to approximate each value to three decimal places , , LAW ENFORCEMENT The formula r L is used by police to estimate the speed r in miles per hour of a car if the length L of the car s skid mark is measures in feet. Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skid mark 00 feet long.. mi/h 0. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h miles above Earth can be approximated by d 000h.What h is the distance to the horizon if a satellite is orbiting 0 miles above Earth? about 00 ft Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

96 Answers (Lesson -) Lesson - - Skills Practice Roots of Real Numbers Use a calculator to approximate each value to three decimal places Simplify (). not a real number y y. s s. x x. a a. m n m n. 00p q 0p q. w v w v. (c) 9c. (a ) b a b Glencoe/McGraw-Hill Glencoe Algebra Answers - Practice (Average) Roots of Real Numbers Use a calculator to approximate each value to three decimal places (0.9) Simplify x 0 9. (a) 0. (a ) not a.. x a real number. 9m t m.. r w. (x) m t r w x. s. m p q 9. x y. x y 9 s pq x y x y 9. m n 0. x y 0. (m ). (x ) m n xy m x. 9a b 0. (x ). d. x0x a b (x ) d x. 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. 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, feet, and 0 feet, what is the area of the garden? Round your answer to the nearest whole number. ft Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

97 Answers (Lesson -) Lesson - - 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 textbook. Suppose the length of a wave is feet. Explain 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 index. a. radicand: index: b. x radicand: x index: c. radicand: index:. 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 x, you must take the absolute value of x 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. Glencoe/McGraw-Hill Glencoe Algebra - Enrichment Approximating Square Roots Consider the following expansion. ( a b a ) a a b b a a a b b a b Think what happens if a is very great in comparison to b. The term small and can be disregarded in an approximation. ( a b a ) a b a is very b a a a b Suppose a number can be expressed as a b, a b. Then an approximate value b of the square root is a a.you should also see that a b a a. b Example Use the formula a b b a to approximate 0 and a. a b. Let a 0 and b. Let a and b. 0 0 ( 0) ( ) Use the formula to find an approximation for each square root to the nearest hundredth. Check your work with a calculator ,, b. Show that a a b a b for a b. a b disregard a b ; a a a b b; a a a b b a a a b ; Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

98 Answers (Lesson -) Lesson - - Study Guide and Intervention Radical Expressions Simplify Radical Expressions For any real numbers a and b, and any integer n : Product Property of Radicals. 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 For any real numbers a and b 0, and any integer n, n a n a b nb, if all roots are defined. 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 exact root. Example Simplify a. b Example a b () a a (b ) b ab a b x Simplify. y x x Quotient Property y y x (x) Factor into squares. (y ) y x (x) Product Property (y ) y x x y y Simplify. x x y Rationalize the y y y denominator. x 0xy Simplify. y Exercises Simplify... a 9 b 0 a b a. x y x y y a... bb pq p a b 9 p q 0 0 Glencoe/McGraw-Hill 9 Glencoe Algebra Answers - Study Guide and Intervention (continued) Radical Expressions Operations with Radicals When you add expressions containing radicals, you can add only like terms or like radical expressions. Two radical expressions are called like radical expressions if both the indices and the radicands are alike. To multiply radicals, use the Product and Quotient Properties. For products of the form (ab cd) (ef gh), use the FOIL method. To rationalize denominators, use conjugates. Numbers of the form ab cd and ab cd, where a, b, c, and d are rational numbers, are called conjugates. The product of conjugates is always a rational number. Example Simplify Factor using squares. 0 Simplify square roots Multiply. 0 0 Combine like radicals. Example Example Simplify ( )( ). ( )( ) 0 Simplify. () () 9 Exercises Simplify ( ). ( 0 ) 9. ( )( ). ( )( ) 9. ( )(0 ) Glencoe/McGraw-Hill 0 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

99 Answers (Lesson -) Lesson - - Simplify. Skills Practice Radical Expressions x 0x x. a b ab. d f f d f.s t s t g z g0gz z. ()(). ()(0) ( )( ) 9. ( )( ) 0. ( )( ). ( )... 0 Glencoe/McGraw-Hill Glencoe Algebra - Simplify. Practice (Average) Radical Expressions t w t w. v z v z z 9. g k gk k 0. x y xy x c d c d 9a d a.. a b 9a b a a. ()(). ()() ( ) 0. ( ). ( )( ). ( 0)( 0) 0 0. ( )( ). ( ). (0 ) x x x x x. 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 expression for s if. Then evaluate s to the nearest mile per hour. 0 ; mi/h. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can be represented by the expressions x y and 9x y. Use the Pythagorean Theorem to find a simplified expression for the measure of the hypotenuse. x y Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

100 Answers (Lesson -) Lesson - - Reading to Learn Mathematics Radical Expressions Pre-Activity How do radical expressions apply to falling objects? Read the introduction to Lesson - at the top of page 0 in your textbook. Describe how you could use the formula given in your textbook 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 expression to be in simplified form. The index 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 expressions used for? to rationalize binomial denominators b. How would you use a conjugate to simplify the radical expression? Multiply numerator and denominator by. c. In order to simplify the radical expression 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 explain it to another person. When rationalizing the denominator in the expression, many students think they should multiply numerator and denominator by. How would you explain 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, which equals. Glencoe/McGraw-Hill Glencoe Algebra Answers - Enrichment Special Products with Radicals Notice that ()(), or (). In general, (x) x when x 0. Also, notice that (9)(). In general, (x)(y) xy when x 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 Example Find the product: ( )( ). ( )( ) () () Example Evaluate ( ). ( ) () () () Multiply.. ( )( ). (0 )(0 ). (x )(x ) x. ( ). (000 0) 0. (y )(y ) y. (0 x) 0 0x x. (x 0) x x 0 You can extend these ideas to patterns for sums and differences of cubes. Study the pattern below. ( x)( x x ) x x Multiply. 9. ( )( 0 ) 0. ( y w)( y yw w ) y w.( 0)( 0 0 ). ( )( ) Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A9 Glencoe Algebra

101 Answers (Lesson -) Lesson - - Study Guide and Intervention Rational Exponents Rational Exponents and Radicals Definition of b n For any real number b and any positive integer n, b n n b, except when b 0 and n is even. m For any nonzero real number b, and any integers m and n, with n, Definition of b m n b m n b n m ( b) n m, except when b 0 and n is even. Example Example Write in radical form. Notice that 0. Evaluate. Notice that 0, 0, and is odd. Exercises Write each expression in radical form Write each radical using rational exponents... a b. p a b p Evaluate each expression (0.000) (0.) Glencoe/McGraw-Hill Glencoe Algebra - Study Guide and Intervention (continued) Rational Exponents Simplify Expressions All the properties of powers from Lesson - apply to rational exponents. When you simplify expressions with rational exponents, leave the exponent in rational form, and write the expression with all positive exponents. Any exponents in the denominator must be positive integers When you simplify radical expressions, you may use rational exponents to simplify, but your answer should be in radical form. Use the smallest index possible. Example Simplify y y. Example y y y y x (x ) Simplify x. ( x ) ( ) ( ) (x ) x x (x) xx Exercises Simplify each expression.. x x. y. p p 0 x y p. m. x x. s m x s 9 p.. a a 9. p x x p a x x x... x a b ab a b b Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A0 Glencoe Algebra

102 Answers (Lesson -) Lesson - - Skills Practice Rational Exponents Write each expression in radical form.... or ( ). (s ) s s Write each radical using rational exponents..... xy x y Evaluate each expression () Simplify each expression.. c c c. m 9 m 9 m 9. q q 0. p p p. x. x x x x x y y y y.. n n n n n.. 9a b a b Glencoe/McGraw-Hill Glencoe Algebra Answers - Practice (Average) Rational Exponents Write each expression in radical form.... m. (n ) or ( ) m or ( m) n n Write each radical using rational exponents m n. a 0 b 9 m n a b Evaluate each expression () Simplify each expression.. g g g 9. s s s 0. u u. y y y b b q q t t t. b. q.. t z z z z z a b ab b 0. ELECTRICITY The amount of current in amperes I that an appliance uses can be P R calculated using the formula I, where P is the power in watts and R is the 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.. amps. BUSINESS A company that produces DVDs uses the formula C n 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. $9 Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

103 Answers (Lesson -) Lesson - - Reading to Learn Mathematics Rational Exponents Pre-Activity How do rational exponents apply to astronomy? Read the introduction to Lesson - at the top of page in your textbook. The formula in the introduction contains the exponent. 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 exponents. n b For any real number b and for any positive integer n, b n except 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, except when b 0 and n is even.. Complete the conditions that must be met in order for an expression with rational exponents to be simplified. It has no negative exponents. It has no fractional exponents in the denominator. It is not a complex fraction. The index of any remaining radical is the number possible. least. Margarita and Pierre were working together on their algebra homework. One exercise asked them to evaluate the expression. Margarita thought that they should raise to the fourth power first and then take the cube root of the result. Pierre thought that they should take the cube root of 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 exponent gives the power and which part gives the root. How can your knowledge of integer exponents help you to keep this straight? Sample answer: An integer exponent can be written as a rational exponent. For example,. 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. Glencoe/McGraw-Hill 9 Glencoe Algebra - Enrichment Lesser-Known Geometric Formulas Many geometric formulas involve radical expressions. 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 and c. c A a c a a A.0 c. The area of an equilateral triangle with a side of length a. Find A when a. A a A. a a a. The area of a regular pentagon with a side of length a. Find A when a. A a 0 a A. a a a a. The area of a regular hexagon with a side of length a. Find A when a 9. A a A 0. a a a a a a. The volume of a regular tetrahedron with an edge of length a. Find V when a. a V a a a V 0.9. The area of the curved surface of a right cone with an altitude of h and radius of base r. Find S when r and h. S rr h S. h a a a. Heron s Formula for the area of a triangle uses the semi-perimeter s, where s a b c. The sides of the triangle have lengths a, b, and c. Find A when a, b, and c. A s(s a)(s b)(s c) A a b. The radius of a circle inscribed in a given triangle also uses the semi-perimeter. Find r when a, b, and c 9. r s(s a)(s b)(s c) r.9 s a b c c Glencoe/McGraw-Hill 0 Glencoe Algebra r r Glencoe/McGraw-Hill A Glencoe Algebra

104 Answers (Lesson -) Lesson - - Study Guide and Intervention 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 Isolate the radical on one side of the equation. Step To eliminate the radical, raise each side of the equation to a power equal to the index of the radical. Step Solve the resulting equation. Step Check your solution in the original equation to make sure that you have not obtained any extraneous roots. Example Solve x. Example x Original equation x Add to each side. x Isolate the radical. x Square each side. x Subtract from each side. x Divide each side by. Check () () The solution x checks. Solve x x. x x Original equation x x x Square each side. x x Simplify. x x Isolate the radical. x x Square each side. x x 0 Subtract x from each side. x(x ) 0 Factor. x 0 or x Check (0), but (0), so 0 is not a solution. (), and (), so the solution is x. Exercises Solve each equation.. x. x. x no solution. x. x. x 0 9 no solution. x 0. 0 x 9. xx x 9. no solution 0. x 0. x x x. 9x x, Glencoe/McGraw-Hill Glencoe Algebra Answers - Study Guide and Intervention (continued) Radical Equations and Inequalities Solve Radical Inequalities A radical inequality is an inequality that has a variable in a radicand. Use the following steps to solve radical inequalities. Step If the index 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. Example Solve 0x. Since the radicand of a square root must be greater than or equal to zero, first solve 0x 0. 0x 0 0x x Now solve 0x. It appears that x is the solution. Test some values. 0x Original inequality 0x Isolate the radical. 0x Eliminate the radical by squaring each side. 0x 0 Subtract from each side. x Divide each side by 0. x x 0 x 0( ) is not a real 0(0), so the 0()., so number, so the inequality is inequality is satisfied. the inequality is not not satisfied. satisfied Therefore the solution x checks. Exercises Solve each inequality.. c. x. 0x 9 c x x. x. x. x x x x 0. 9 x. 9. x x x x x 0. x. d d. b b 0 x 0 d b Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

105 Answers (Lesson -) Lesson - - Skills Practice Radical Equations and Inequalities Solve each equation or inequality.. x. x. j. v 0 no solution. y no solution. w. b. n 9. r 0. p. k 0. (d ). (t ). ( u) 0 9. z z no solution. g g. x x no solution. s s 9. x x 0. a a. r 0 r. x x 0. y y. r r Glencoe/McGraw-Hill Glencoe Algebra - Practice (Average) Radical Equations and Inequalities Solve each equation or inequality.. x. x 9. p 0. h 0. c 9 9. h no solution. d. w 9. q 9 0. y 9 0 no solution. m 0. m. n. t. t. (v ) no solution. (g ). (u ) 0 9. d d 0. r r. x x 0. x x no solution. a a. z z. q no solution. a a. 9 c c. x x 9. 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 Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

106 Answers (Lesson -) Lesson - - Reading to Learn Mathematics Radical Equations and Inequalities Pre-Activity How do radical equations apply to manufacturing? Read the introduction to Lesson - at the top of page in your textbook. Explain how you would use the formula in your textbook 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 extraneous 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 b. Describe two ways you can check the proposed solutions of a radical equation in order to determine whether any of them are extraneous 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 extraneous.. Complete the steps that should be followed in order to solve a radical inequality. Step If the index 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 explain 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 explain 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 example, the only solution of x is, but the squared equation x has two solutions, and. Glencoe/McGraw-Hill Glencoe Algebra Answers - Enrichment 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 complex statement is true. Example Under what conditions is P Q true? Create the truth table for the statement. Use the information from the truth table above for P Q to complete the last column. P Q P P Q T T F T When one statement is T F F F true and one is false, F T T T the conjunction is true. F F T T The truth table indicates that P Q is true in all cases except where P is true and Q is false. Use truth tables to determine the conditions under which each statement is true.. P Q. P (P Q) all except where both all P and Q are true. (P Q) (P Q). (P Q) (Q P) all all. (P Q) (Q P). (P Q) (P Q) both P and Q are true; all both P and Q are false Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

107 Answers (Lesson -9) Lesson -9-9 Study Guide and Intervention Complex Numbers Add and Subtract Complex Numbers Acomplex number is any number that can be written in the form a bi, Complex 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 Complex Numbers (a bi) (c di) (a c) (b d )i Example Simplify ( i) ( i). Example ( i) ( i) ( ) ( )i 0 i ( i) ( i) ( ) [ ()]i i Simplify ( i) ( i). To solve a quadratic equation that does not have real solutions, you can use the fact that i to find complex solutions. Example Solve x 0. x 0 Original equation x Subtract from each side. x Divide each side by. x Take the square root of each side. x i Exercises Simplify.. ( i) ( i). ( i) ( i). ( i) ( i) i i 0 i. ( i) ( i). ( i) ( i). ( i) ( i) 9i i. ( i) ( i). (9 i) ( i) 9. ( i) ( i) i i i 0. i. i. i i Solve each equation.. x 0. x 0. 9x 9 i i i Glencoe/McGraw-Hill Glencoe Algebra -9 Study Guide and Intervention (continued) Complex Numbers Multiply and Divide Complex Numbers Multiplication of Complex Numbers Use the definition of i and the FOIL method: (a bi)(c di) (ac bd ) (ad bc)i To divide by a complex number, first multiply the dividend and divisor by the complex conjugate of the divisor. Complex Conjugate a bi and a bi are complex conjugates. The product of complex conjugates is always a real number. Example Simplify ( i) ( i). ( i) ( i) () (i) (i)() (i)(i) FOIL i 0i 0i Multiply. i 0() Simplify. i Standard form Example i i i Simplify. i i i i i Use the complex conjugate of the divisor. 9i i i Multiply. 9i i i i Standard form Exercises Simplify.. ( i)( i) i. ( i)( i) i. ( i)( i) 0i. ( i)( i). ( i)( i) i. ( i)( i) i. ( i)( i)( i). ( i)( i)( i) 9. ( i)( i)( i) i i i i i 0. i. i. i i i i i i i. i. i. i i i i i i i.. i. i i i i i Glencoe/McGraw-Hill Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

108 Answers (Lesson -9) Lesson -9-9 Simplify. Skills Practice Complex Numbers. i. 9 i. x 9 x i.. (i)(i)(i) 0i. i i. i i. ( i) ( i) i 9. ( i) ( i) i 0. (0 i) ( i) i. ( i)( i) i. ( i)( i) i. ( i)( i) i. ( i)( i) i i i.. i i i 0 Solve each equation.. x 0 i. x 0 i 9. x 0 0 i 0. x 0 i. x 0 i. x 9 0 i Find the values of m and n that make each equation true.. 0 i m ni,. m i ni,. ( m) ni 9 i,. ( n) (m )i i, Glencoe/McGraw-Hill 9 Glencoe Algebra Answers -9 Simplify. Practice (Average) Complex Numbers. 9 i. i. s s i. a b.. a b ia. (i)(i)(i). (i) (i) 9. i 0i 9i 0. i. i 9. ( i) ( i) i i 0i. ( i) (9 i). ( i) ( i). (0 i) ( 0i) i 9i i. ( i) (0 0i). ( i)( i). ( i)( i) i i 9. ( i)( i) 0. ( i)(9 i). i i i i i i i i i... i i i i Solve each equation.. n 0 i. m 0 0 i. m 0 i9. m 0 i 9. m 0 i 0. x 0 i Find the values of m and n that make each equation true.. i m ni,. ( m) ni i, 9. (m ) ( n)i i,. ( n) (m 0)i i,. ELECTRICITY The impedance in one part of a series circuit is j ohms and the impedance in another part of the circuit is j ohms. Add these complex numbers to find the total impedance in the circuit. j ohms. 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 Glencoe/McGraw-Hill 90 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra

109 Answers (Lesson -9) Lesson -9-9 Reading to Learn Mathematics Complex Numbers Pre-Activity How do complex numbers apply to polynomial equations? Read the introduction to Lesson -9 at the top of page 0 in your textbook. 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 complex number. b. In the complex number i, the real part is and the imaginary part is. This is an example of a complex number that is also a(n) imaginary number. c. In the complex number, the real part is and the imaginary part is 0. This is example of complex number that is also a(n) real number. d. In the complex number i, the real part is 0 and the imaginary part is. This is an example of a complex number that is also a(n) pure imaginary number.. Give the complex conjugate of each number. a. i b. i i i. Why are complex conjugates used in dividing complex numbers? The product of complex conjugates is always a real number.. Explain how you would use complex conjugates to find ( i) ( 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 expression 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. Glencoe/McGraw-Hill 9 Glencoe Algebra -9 Enrichment Conjugates and Absolute Value When studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z x yi.we denote the conjugate of z by z. Thus, z x yi. We can define the absolute value of a complex number as follows. z x yi x y There are many important relationships involving conjugates and absolute values of complex numbers. Example Let z x yi. Then, z (x yi)(x yi) x y Show z zz for any complex number z. (x ) y z Example z Show is the multiplicative inverse for any nonzero z complex number z. z. z z Thus, is the multiplicative inverse of z. z We know z zz. If z 0, then we have z For each of the following complex numbers, find the absolute value and multiplicative inverse.. i ; i. i ; i. i ; 9 i. i ; i. i ; 9 i. i ; i. i. i 9. i i ; ; i ; i Glencoe/McGraw-Hill 9 Glencoe Algebra Glencoe/McGraw-Hill A Glencoe Algebra