Factoring Chapter Overview and Pacing

Transcription

1 Factoring Chapter Overview and Pacing PACING (days) Regular Block Basic/ Basic/ Average Advanced Average Advanced Factors and Greatest Common Factors (pp ) Find prime factorizations of integers and monomials. Find the greatest common factors of integers and monomials. LESSON OBJECTIVES Year-long and two-year pacing: pages T0 T. Factoring Using the Distributive Property (pp ) 0.5 Preview: Use algebra tiles and a product mat to factor binomials. (with 9- Factor polynomials by using the Distributive Property. Preview) Solve quadratic equations of the form a b 0. Factoring Trinomials: b c (pp ) 3 Preview: Use algebra tiles to factor trinomials. (with 9-3 (with 9-3 Factor trinomials of the form b c. Preview) Preview) Solve equations of the form b c 0. Factoring Trinomials: a b c (pp ) Factor trinomials of the form a b c. Solve equations of the form a b c 0. Factoring Differences of Squares (pp ) Factor binomials that are the differences of squares. Solve equations involving the differences of squares. Perfect Squares and Factoring (pp ) Factor perfect square trinomials. Solve equations involving perfect squares. Study Guide and Practice Test (pp ) 0.5 Standardized Test Practice (pp. 50 5) Chapter Assessment TOTAL An electronic version of this chapter is available on StudentWorks TM. This backpack solution CD-ROM allows students instant access to the Student Edition, lesson worksheet pages, and web resources. 47A Chapter 9 Factoring

2 Timesaving Tools Chapter Resource Manager All-In-One Planner and Resource Center See pages T5 and T. Study Guide and Intervention CHAPTER 9 RESOURCE MASTERS Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Prerequisite Skills Workbook Applications* Parent and Student Study Guide Workbook 5-Minute Check Transparencies Interactive Chalkboard AlgePASS: Tutorial Plus (lessons) Materials SC (Preview: algebra tiles, product mat) , , 5 (Preview: algebra tiles, product mat), graphing calculator GCS GCS 40, straightedge, scissors, SC 8, graph paper SM , *Key to Abbreviations: GCS Graphing Calculator and Spreadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 9 Factoring 47B

3 Mathematical Connections and Background Continuity of Instruction Prior Knowledge Students studied prime numbers and greatest common factors in previous courses. They also found the prime factorization of numbers. In Chapter 8, students learned the rules for dividing monomials. This Chapter This chapter covers the factoring of integers, monomials and polynomials. Students learn to find the greatest common factor of the monomials in a polynomial. They use the GCF and the Distributive Property to factor polynomials. Students also apply the Zero Product Property to solve quadratic equations. Future Connections Factoring polynomials is used to solve many real-world problems. It is basic to studying more about polynomial equations and functions. Factors and Greatest Common Factors The factors of a given number are all the numbers that divide the number evenly. This includes the number itself and. The factors of a number can be found by determining all the pairs of numbers whose product is that number. Natural numbers greater than are classified as either prime or composite. Prime numbers have eactly two factors while composite numbers have more than two factors. The number is neither prime nor composite. A prime factorization is the epression of a number as the product of factors that are all prime numbers. The prime factorization of a negative integer is epressed as the product and prime numbers. Monomials can be written in factored form. A monomial in factored form is the product of prime numbers and variables. Variables, however, cannot have an eponent greater than. So 3 must be written as in factored form. Prime factorizations are used to determine the greatest common factor (GCF) of two or more integers or monomials. The GCF is the product of all the common prime factors of the two integers or monomials. If is the only common factor, then they are relatively prime. Factoring Using the Distributive Property Many polynomials also have factors. Some polynomials are the product of a polynomial and a monomial. Reverse the process of multiplying a polynomial by a monomial to factor using the Distributive Property. First find the greatest common factor of the terms of the polynomial. If the GCF is not, then rewrite each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. If a polynomial has four or more terms you can factor by grouping. Group terms in pairs that have common factors. Use the Distributive Property to factor the GCF from each pair of terms. The binomials in each pair of factored terms should be identical. Use the Distributive Property to factor out the common binomial factor. The remaining factors are grouped to form a second binomial. It may appear that the factored pairs do not have identical binomials, but one may be the additive inverse of the other. Write one as the product of and its additive inverse. Then multiply the GCF of that pair by. If the product of two factors is 0, then at least one of them is 0 according to the Zero Product Property. If an equation has the form ab 0 or can be written in this form by factoring, then the Zero Product Property can be applied to solve the equation. Set each factor equal to 0 and solve each resulting equation. 47C Chapter 9 Factoring

4 Factoring Trinomials: b c The FOIL method was used to multiply two binomials. Reverse the FOIL method to factor a quadratic polynomial of the form b c into two binomials. Find two numbers m and n whose product is c and whose sum is b. The two numbers are the last terms of the two binomials ( m) and ( n). If b is negative and c is positive then m and n must both be negative. If c is negative, then m and n must have different signs. This is because the product of two numbers with different signs is negative. The Zero Product Property can be used to solve some quadratic equations written in the form b c 0. Factor the trinomial, and then set each factor equal to 0. Solve each equation to find the solution of the quadratic equation. Be sure to check the solutions in the original equation. Factoring Trinomials: a b c To factor a trinomial in which the coefficient of is not, first check to see if the terms of the polynomial have a GCF. If so, factor it out. If the coefficient of is still not, or there is no GCF, factor a b c by making an organized list of the factors of the product of a and c. For eample, to factor 8 5 3, make an organized list of the factors of 8 ( 3), or 4. Look for a pair of factors, m and n, whose sum is equal to b in the trinomial. Then rewrite the trinomial, replacing b with m n. The new polynomial has four terms. Use the factoring by grouping technique shown in Lesson 9- to factor this polynomial into two binomial factors. A polynomial that cannot be factored is a prime polynomial. Use the above method and the Zero Product Property to solve equations in the form a b c 0. Factoring Differences of Squares Lesson 8-8 discussed the pattern for the product of a sum and a difference: (a b)(a b) a b. The binomial a b is called the difference of two squares and can be factored as the product of a sum and a difference: a b (a b)(a b). To do this, identify a and b, the square roots of the first and last terms, respectively. Then apply the pattern. Other aspects of factoring to watch for are factoring out a common factor, applying a technique more than once, and applying several techniques. Use any of the appropriate techniques and the Zero Product Property to solve many polynomial equations. Perfect Squares and Factoring Some trinomials have patterns that make their factoring easier. In Lesson 8-8, students learned about patterns for the square of a sum, (a b) a ab b, and the square of a difference, (a b) a ab b. These products, a ab b and a ab b, are called perfect square trinomials, because they are the result of squaring a binomial. To recognize a perfect square trinomial, first determine if the first and last terms are perfect squares. Then find the square roots of the first and last terms, checking to see if twice the product of these square roots is equal to the middle term of the trinomial. If the trinomial is a perfect square, and the middle term is positive use the pattern a ab b (a b) to factor it. If the middle term is negative, use the pattern a ab b (a b). It is important to note that the last term of a perfect square trinomial cannot be negative. If one side of an equation is a perfect square or can be written as a perfect square, then the Square Root Property can be applied to solve the equation. The Square Root Property allows you to take the square root of each side of an equation, so long as both the positive and negative square roots of a number are taken into account. So, for any number n that is greater than 0, if n, then n. Two solutions result from such equations, one using the positive square root and one using the negative square root. Quick Review Math Handbook Hot Words includes a glossary of terms while Hot Topics consists of eplanations of key mathematical concepts with eercises to test comprehension. This valuable resource can be used as a reference in the classroom or for home study. Lesson Hot Topics Section Lesson Hot Topics Section GS P GS = Getting Started, P = Preview Additional mathematical information and teaching notes are available at Chapter 9 Factoring 47D

5 and Assessment Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Type Student Edition Teacher Resources Technology/Internet ASSESSMENT INTERVENTION Ongoing Prerequisite Skills, pp. 473, 479, 486, 494, 500, 506 Practice Quiz, p. 486 Practice Quiz, p. 500 Mied Review Error Analysis Standardized Test Practice Open-Ended Assessment Chapter Assessment pp. 479, 486, 494, 500, 506, 54 Find the Error, pp. 49, 498, 504 Common Misconceptions, pp. 483, 50 pp. 479, 486, 494, 500, 503, 505, 54, 59, 50 5 Writing in Math, pp. 479, 485, 494, 500, 506, 54 Open Ended, pp. 477, 484, 49, 498, 504, 5 Standardized Test, p. 5 Study Guide, pp Practice Test, p Minute Check Transparencies Prerequisite Skills Workbook, pp. 3 4 Quizzes, CRM pp Mid-Chapter Test, CRM p. 575 Study Guide and Intervention, CRM pp , , , 54 54, , Cumulative Review, CRM p. 576 Find the Error, TWE pp. 49, 498, 504 Unlocking Misconceptions, TWE p. 497 Tips for New Teachers, TWE pp. 490, 509 TWE pp Standardized Test Practice, CRM pp Modeling: TWE pp. 479, 506 Speaking: TWE pp. 486, 494 Writing: TWE pp. 500, 54 Open-Ended Assessment, CRM p. 57 Multiple-Choice Tests (Forms, A, B), CRM pp Free-Response Tests (Forms C, D, 3), CRM pp Vocabulary Test/Review, CRM p. 57 AlgePASS: Tutorial Plus, Lessons 4, 5, 6, and Standardized Test Practice CD-ROM standardized_test EamView Pro (see below) MindJogger Videoquizzes vocabulary_review For more information on Yearly ProgressPro, see p Algebra Lesson Yearly ProgressPro Skill Lesson 9- Factors and Greatest Common Factors 9- Factoring Using the Distributive Property 9-3 Factoring Trinomials b c 9-4 Factoring Trinomials a b c 9-5 Factoring Differences of Squares 9-6 Perfect Squares and Factoring EamView Pro Use the networkable EamView Pro to: Create multiple versions of tests. Create modified tests for Inclusion students. Edit eisting questions and add your own questions. Use built-in state curriculum correlations to create tests aligned with state standards. Change English tests to Spanish and vice versa. For more information on Intervention and Assessment, see pp. T8 T. 47E Chapter 9 Factoring

6 Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Study Organizer, p. 473 Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 477, 484, 49, 498, 504, 5) Reading Mathematics, p. 507 Writing in Math questions in every lesson, pp. 479, 485, 494, 500, 506, 54 Reading Study Tip, pp. 489, 5 WebQuest, p. 479 Teacher Wraparound Edition Foldables Study Organizer, pp. 473, 55 Study Notebook suggestions, pp. 477, 480, 484, 488, 49, 498, 504, 507, 5 Modeling activities, pp. 479, 506 Speaking activities, pp. 486, 494 Writing activities, pp. 500, 54 Differentiated Instruction, (Verbal/Linguistic), p. 475 ELL Resources, pp. 47, 475, 478, 485, 493, 499, 505, 507, 53, 55 Additional Resources Vocabulary Builder worksheets require students to define and give eamples for key vocabulary terms as they progress through the chapter. (Chapter 9 Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter 9 Resource Masters,pp. 57, 533, 539, 545, 55, 559) Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Lesson 9- Building on Prior Knowledge Have groups of students find the greatest common factor (GCF) of two numbers. For eample, find the GCF of 54 and Circle common prime factors. So, Then, show students how to find the GCF of a set of monomials. Have students note similarities in the procedures. Lesson 9-3 Fleible Groups Give groups of four students a set of four cards with one trinomial on each card. Each group should receive a trinomial where b and c are positive, b is positive and c is negative, b is negative and c is positive, and b and c are negative. Have groups develop rules of how to find the factors of b and c and how to ultimately find the factors of the trinomial. Allow groups to share their findings with other groups. Lesson 9-6 Peer Tutoring Give students a set of trinomials. You can use Eercises 7 on page 5 or make up your own. Have students work with a partner to analyze each problem and state whether it is a perfect square trinomial. Students can then factor each trinomial and find a solution set. Chapter 9 Factoring 47F

7 Notes Factoring Have students read over the list of objectives and make a list of any words with which they are not familiar. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson. Lesson 9- Find the prime factorizations of integers and monomials. Lesson 9- Find the greatest common factors (GCF) for sets of integers and monomials. Lessons 9- through 9-6 Factor polynomials. Lessons 9- through 9-6 Use the Zero Product Property to solve equations. Key Vocabulary The factoring of polynomials can be used to solve a variety of real-world problems and lays the foundation for the further study of polynomial equations. Factoring is used to solve problems involving vertical motion. For eample, the height h in feet of a dolphin that jumps out of the water traveling at 0 feet per second is modeled by a polynomial equation. Factoring can be used to determine how long the dolphin is in the air. You will learn how to solve polynomial equations in Lesson 9-. factored form (p. 475) factoring by grouping (p. 48) prime polynomial (p. 497) difference of squares (p. 50) perfect square trinomials (p. 508) NCTM Local Lesson Standards Objectives 9-,, 6, 8, 9, 0 9-, 6, 8, 0 Preview 9-, 6, 8, 9, 0 9-3, 6, 8, 0 Preview 9-3, 6, 8, 9, 0 9-4, 6, 8, 9, 0 9-5, 3, 6, 8, 9, 0 9-6, 6, 8, 9, 0 Key to NCTM Standards: =Number & Operations, =Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 0=Representation 47 Chapter 9 Factoring Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 9 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 9 test. 47 Chapter 9 Factoring

8 Prerequisite Skills To be successful in this chapter, you ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 9. For Lessons 9- through 9-6 Distributive Property Rewrite each epression using the Distributive Property. Then simplify. (For review, see Lesson -5.). 3(4 ) 3. a(a 5) a 5a 3. 7(n 3n ) 4. 6y( 3y 5y y 3 ) 7n n 7 8y 30y 3 6y 4 For Lessons 9-3 and 9-4 Multiplying Binomials Find each product. (For review, see Lesson 8-7.) 5. ( 4)( 7) 6. (3n 4)(n 5) 7. (6a b)(9a b) 8. ( 8y)( y) 8 3n n 0 54a ab b 4y 96y For Lessons 9-5 and 9-6 Special Products Find each product. (For review, see Lesson 8-8.) 9. (y 9) 0. (3a ). (n 5)(n 5). (6p 7q)(6p 7q) y 8y 8 9a a 4 n 5 36p 49q For Lesson 9-6 Square Roots Find each square root. (For review, see Lesson -7.) This section provides a review of the basic concepts needed before beginning Chapter 9. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp Prerequisite Skills in the Getting Ready for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Prerequisite Lesson Skill 9- Distributive Property, p Multiplying Polynomials, p Factoring by Grouping, p Square Roots, p Special Products, p. 506 Factoring Make this Foldable to help you organize your notes. Begin with a sheet of plain 8 " by " paper. Fold in Siths Fold Again Fold in thirds and then in half along the width. Open. Fold lengthwise, leaving a " tab on the right. Cut Open. Cut the short side along the folds to make tabs. Label each tab as shown. Label F a c t o r i n g Reading and Writing As you read and study the chapter, write notes and eamples for each lesson under its tab. Chapter 9 Factoring 473 For more information about Foldables, see Teaching Mathematics with Foldables. TM Organization of Data and Questioning Before beginning each lesson, ask students to think of one question that comes to mind as they skim through the lesson. Write the question on the front of the corresponding lesson tab. As students read and work through the lesson, ask them to record the answer to their question under the tab. Students can also use their Foldables to take notes, record concepts, define terms, and record other questions that arise about factoring. Chapter 9 Factoring 473

9 Lesson Notes Factors and Greatest Common Factors Focus 5-Minute Check Transparency 9- Use as a quiz or review of Chapter 8. Mathematical Background notes are available for this lesson on p. 47C. are prime numbers related to the search for etraterrestrial life? Ask students: What is a prime number? A prime number is any whole number, greater than one, whose only factors are one and itself. Why might a radio signal from space composed of prime numbers be significant? Sample answer: A signal composed of only prime numbers would seem to signify that it was sent by intelligent beings. Vocabulary prime number composite number prime factorization factored form greatest common factor (GCF) Virginia SOL STANDARD A. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. TEACHING TIP An alternative definition is that a prime number is a positive integer with eactly two different factors. You may want to point out that is the only even prime number. Find prime factorizations of integers and monomials. Find the greatest common factors of integers and monomials. are prime numbers related to the search for etraterrestrial life? In the search for etraterrestrial life, scientists listen to radio signals coming from faraway galaies. How can they be sure that a particular radio signal was deliberately sent by intelligent beings instead of coming from some natural phenomenon? What if that signal began with a series of beeps in a pattern comprised of the first 30 prime numbers ( beep-beep, beep-beep-beep, and so on)? PRIME FACTORIZATION Recall that when two or more numbers are multiplied, each number is a factor of the product. Some numbers, like 8, can be epressed as the product of different pairs of whole numbers. This can be shown geometrically. Consider all of the possible rectangles with whole number dimensions that have areas of 8 square units The number 8 has 6 factors,,, 3, 6, 9, and 8. Whole numbers greater than can be classified by their number of factors. Words A whole number, greater than, whose only factors are and itself, is called a prime number. A whole number, greater than, that has more than two factors is called a composite number. Prime and Composite Numbers Eamples, 3, 5, 7,, 3, 7, 9 4, 6, 8, 9, 0,, 4, 5, 6, 8 Resource Manager Study Tip Listing Factors Notice that in Eample, 6 is listed as a factor of 36 only once. 474 Chapter 9 Factoring Eample 0 and are neither prime nor composite. Classify Numbers as Prime or Composite Factor each number. Then classify each number as prime or composite. a. 36 To find the factors of 36, list all pairs of whole numbers whose product is Therefore, the factors of 36, in increasing order, are,, 3, 4, 6, 9,, 8, and 36. Since 36 has more than two factors, it is a composite number. Workbook and Reproducible Masters Chapter 9 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 55 Practice, p. 56 Reading to Learn Mathematics, p. 57 Enrichment, p. 58 Parent and Student Study Guide Workbook, p. 68 Prerequisite Skills Workbook, pp. 3 4 Transparencies 5-Minute Check Transparency 9- Answer Key Transparencies Technology Interactive Chalkboard

10 Study Tip Prime Numbers Before deciding that a number is prime, try dividing it by all of the prime numbers that are less than the square root of that number. TEACHING TIP Point out that students could have started with 45, 3 30, or 5 8, and found the same prime factorization. Study Tip Unique Factorization Theorem The prime factorization of every number is unique ecept for the order in which the factors are written. b. 3 The only whole numbers that can be multiplied together to get 3 are and 3. Therefore, the factors of 3 are and 3. Since the only factors of 3 are and itself, 3 is a prime number. When a whole number is epressed as the product of factors that are all prime numbers, the epression is called the prime factorization of the number. Eample Prime Factorization of a Positive Integer Find the prime factorization of 90. Method The least prime factor of 90 is. 3 5 The least prime factor of 45 is The least prime factor of 5 is 3. All of the factors in the last row are prime. Thus, the prime factorization of 90 is Method Use a factor tree and 0 5 All of the factors in the last branch of the factor tree are prime. Thus, the prime factorization of 90 is or 3 5. Usually the factors are ordered from the least prime factor to the greatest. Anegative integer is factored completely when it is epressed as the product of and prime numbers. Eample 3 Prime Factorization of a Negative Integer Find the prime factorization of Epress 40 as times Thus, the prime factorization of 40 is 5 7 or 5 7. Teach PRIME FACTORIZATION In-Class Eamples 3 Power Point Factor each number. Then classify each number as prime or composite. a. The factors are,,, and, so the number is composite. b. 3 The factors are and 3, so the number is prime. Teaching Tip Eplain that the prime factorization for each number is unique, so as long as you divide by prime numbers, you will get the same prime factorization. For eample: , which is the same as that was found in Eample. Find the prime factorization of or 3 7 Find the prime factorization of 3. 3 Concept Check Prime Factorization José and Latecia both found the prime factorization of 60. José got 3 5, and Latecia got 3 5. Eplain which is correct. Both are correct. Each number has a unique prime factorization. Amonomial is in factored form when it is epressed as the product of prime numbers and variables and no variable has an eponent greater than. Lesson 9- Factors and Greatest Common Factors 475 Differentiated Instruction ELL Verbal/Linguistic Factoring integers is a very visual skill, whether it is done by writing out factors in a line, or by using a factor tree. Have students describe one of the methods of factoring in their own words, without actually writing out the factors as an eample. Alternatively, have students write a description of how to factor, using an eample as a guide. Lesson 9- Factors and Greatest Common Factors 475

11 In-Class Eample 4 Factor each monomial completely. a. 8 3 y 3 3 y y y b. 6rst 3 r s t t In-Class Eamples 6 Teaching Tip Remind students that prime factorization of a constant can have eponents greater than, but prime factorization of variable values cannot. GREATEST COMMON FACTOR 5 Find the GCF of each set of monomials. a. and 8 The GCF is 6 b. 7a b and 5ab c The GCF is 3ab CRAFTS Rene has crocheted 3 squares for an afghan. Each square is foot square. She is not sure how she will arrange the squares but does know it will be rectangular and have a ribbon trim. What is the maimum amount of ribbon she might need to finish the afghan? 66 ft Interactive Chalkboard PowerPoint Presentations Power Point Power Point This CD-ROM is a customizable Microsoft PowerPoint presentation that includes: Step-by-step, dynamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The 5-Minute Check Transparencies Hot links to Glencoe Online Study Tools Study Tip Alternative Method You can also find the greatest common factor by listing the factors of each number and finding which of the common factors is the greatest. Consider Eample 5a. 5:, 3, 5, 5 6:,, 4, 8, 6 The only common factor, and therefore, the greatest common factor, is. 476 Chapter 9 Factoring Eample 4 Prime Factorization of a Monomial Factor each monomial completely. a. a b 3 a b 3 6 a a b b b 3 a a b b b 6 3 6, a a a, and b 3 b b b Thus, a b 3 in factored form is 3 a a b b b. b. 66pq 66pq 66 p q q Epress 66 as times p q q p q q 33 3 Thus, 66pq in factored form is 3 p q q. GREATEST COMMON FACTOR Two or more numbers may have some common prime factors. Consider the prime factorization of 48 and Factor each number Circle the common prime factors. The integers 48 and 60 have two s and one 3 as common prime factors. The product of these common prime factors, 3 or, is called the greatest common factor (GCF) of 48 and 60. The GCF is the greatest number that is a factor of both original numbers. Greatest Common Factor (GCF) The GCF of two or more integers is the product of the prime factors common to the integers. The GCF of two or more monomials is the product of their common factors when each monomial is in factored form. If two or more integers or monomials have a GCF of, then the integers or monomials are said to be relatively prime. Eample 5 GCF of a Set of Monomials Find the GCF of each set of monomials. a. 5 and Factor each number. 6 Circle the common prime factors, if any. There are no common prime factors, so the GCF of 5 and 6 is. This means that 5 and 6 are relatively prime. b. 36 y and 54y z 36 y 3 3 y 54y z y y z Factor each number. The GCF of 36 y and 54y z is 3 3 y or 8y. Teacher to Teacher Circle the common prime factors. Lisa Cook Kaysville Jr. H.S., Kaysville, UT To help in identifying prime factors, I like to have my students eplore the Sieve of Eratosthenes. Use a 0-by-0 grid with the numbers -00 on it. Cross out since prime numbers are greater than. Circle and cross out all multiples of. Circle 3 and cross out multiples of 3. Continue with the net odd number until all multiples have been eliminated. The circled numbers are the prime numbers less than 00." 476 Chapter 9 Factoring

12 Concept Check Guided Practice GUIDED PRACTICE KEY Eercises Eamples Eample 6 Practice and Apply Use Factors GEOMETRY The area of a rectangle is 8 square inches. If the length and width are both whole numbers, what is the maimum perimeter of the rectangle? Find the factors of 8, and draw rectangles with each length and width. Then find each perimeter. The factors of 8 are,, 4, 7, 4, and 8. 4 P 4 4 or 3 8 P 8 8 or 58 The greatest perimeter is 58 inches. The rectangle with this perimeter has a length of 8 inches and a width of inch.. Determine whether the following statement is true or false. If false, provide a countereample. All prime numbers are odd. false;. Eplain what it means for two numbers to be relatively prime. Their GCF is. 3. OPEN ENDED Name two monomials whose GCF is 5. Sample answer: 5 and 0 3 Find the factors of each number. Then classify each number as prime or composite. 4. 8,, 4, 8; composite 5. 7, 7; prime 6.,, 4, 7, 8, 4, 6, 8, 56, ; Find the prime factorization of each integer. composite Lesson 9- Factors and Greatest Common Factors P or , Factor each monomial completely p p p. 39b 3 c yz 3 3 b b b c c. 5 5 Find the GCF of each set of monomials. y z z 3. 0, y, 36y 8y 5. 54, 63, n, m 7. a b, 90a b c 6a b 8. 5r, 35s, 70rs 5 (relatively prime) Application 9. GARDENING Ashley is planting 0 tomato plants in her garden. In what ways 9. 5 rows of 4 plants, 6rows of 0 plants, can she arrange them so that she has the same number of plants in each row, at least 5 rows of plants, and at least 5 plants in each row? 8 rows of 5 plants, 0 rows of plants, rows of 0 plants, 5 rows of 8 plants, 0 rows of 6 plants or 4 rows of 5 plants indicates increased difficulty 0 7. See margin. Find the factors of each number. Then classify each number as prime or composite Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 9. include eplanations on how to factor numbers, find prime factorizations, and find the greatest common factor. include any other item(s) that they find helpful in mastering the skills in this lesson. About the Eercises Organization by Objective Prime Factorization: 0 7, 3 39 Greatest Common Factor: 48 6 Odd/Even Assignments Eercises 0 9 and 3 6 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 9 odd, 30, 3, 33 6 odd, Average: 9 odd, 33 6 odd, 63 66, Advanced: 0 8 even, 3 6 even, (optional: 8 86) Answers 0., 9; prime., 5, 5; composite.,, 4, 5, 8, 0, 6, 0, 40, 80; composite 3., 6; prime 4., 7, 3, 9; composite 5., 7, 7, 9; composite 6.,, 3, 6, 7, 9, 4, 8,, 4, 63, 6; composite 7.,, 4, 8, 6, 9, 38, 76, 5, 304; composite Lesson 9- Factors and Greatest Common Factors 477

13 Study Study Guide and and Intervention Intervention, p. 53 (shown) and p NAME DATE PERIOD Factors and Greatest Common Factors Prime Factorization When two or more numbers are multiplied, each number is called a factor of the product. Prime Number Definition Aprime number is a whole number, greater than, whose only factors are and itself. Eample Acomposite number is a whole number, greater than, that has more than Composite Number 0 two factors. Prime Factorization Prime factorization occurs when a whole number is epressed as a product of factors that are all prime numbers. Eample Factor each number. Eample Find the prime Then classify each number as prime or factorization of 00. composite. Method a To find the factors of 8, list all pairs of 50 whole numbers whose product is Therefore, the factors of 8 are,, 4, 7, All the factors in the last row are prime, so 4, and 8. Since 8 has more than the prime factorization of 00 is 3 5. factors, it is a composite number. Method b. 3 Use a factor tree. 00 To find the factors of 3, list all pairs of 00 whole numbers whose product is Therefore, the factors of 3 are and Since the only factors of 3 are itself and All of the factors in each last branch of the, it is a prime number. factor tree are prime, so the prime factorization of 00 is 3 5. Eercises Find the factors of each number. Then classify the number as prime or composite.. 4, 4; prime.,, ; composite 3. 90,, 3, 5, 6, 9, 0, 5, 8, 30, , 3, 5, 5, 9, 573, 955, 865; 45, 90; composite composite Find the prime factorization of each integer Factor each monomial completely m n 0. 49a 3 b 3 3 m m n 7 7 a a a b b Skills Practice Practice, p. 55 and (Average) Practice, p. 56 (shown) 9- NAME DATE PERIOD Factors and Greatest Common Factors Find the factors of each number. Then classify each number as prime or composite.. 8,, 3, 6, 9, 8;. 37, 37; prime 3. 48,, 3, 4, 6, 8,, composite 6, 4, 48; composite 4. 6,, 4, 9, 58, 6; 5. 38,, 3, 6, 3, 46, 6., ; prime composite 69, 38; composite Find the prime factorization of each integer Factor each monomial completely d mn 3 5 d d d d d 3 3 m n 5 Lesson 9- For Eercises See Eamples 0 7, 6, 65, , , 5 63, , 67 6 Etra Practice See page 839. GEOMETRY For Eercises 8 and 9, consider a rectangle whose area is 96 square millimeters and whose length and width are both whole numbers. 8. What is the minimum perimeter of the rectangle? Eplain your reasoning. 40 mm 9. What is the maimum perimeter of the rectangle? Eplain your reasoning. 94 mm 8 9. See margin for eplanations. COOKIES For Eercises 30 and 3, use the following information. A bakery packages cookies in two sizes of boes, one with 8 cookies and the other with 4 cookies. A small number of cookies are to be wrapped in cellophane before they are placed in a bo. To save money, the bakery will use the same size cellophane packages for each bo. 30. How many cookies should the bakery place in each cellophane package to maimize the number of cookies in each package? 6 cookies 3. How many cellophane packages will go in each size bo? 3 packages in the bo of 8 cookies and 4 packages in the bo of 4 cookies. Find the prime factorization of each integer Factor each monomial completely See margin d y 4. 49a 3 b gh 44. 8pq n 3 m yz a bc Find the GCF of each set of monomials , , , , , 0, , 63, a, 8b 55. 4d, 30c d 6d 56. 0gh, 36g h 4gh 57. p q, 3r t 58. 8, 30y, 54y a, 63a 3 b, 9b m n, 8mn, m n 3 mn 6. 80a b, 96a b 3, 8a b 6a b 5. 8b c abc b b c c c 5 9 a b c c c 7. 68pq r 8. yz 3 7 p q q r y z z Find the GCF of each set of monomials. 9. 8, , 45, , 4, NUMBER THEORY Twin primes are two consecutive odd numbers that are prime. The first pair of twin primes is 3 and 5. List the net five pairs of twin primes. 5, 7;, 3; 7, 9; 9, 3; 4, 43., 30, , 7, , 7, fg 5,56f 3 g 8fg 6. 7r s,36rs 3 36rs 7. 5a b,35ab 5ab 8. 8m 3 n,45pq 9. 40y,56 3 y, 4 y 3 4y c 3 d,40c d,3c d 8c d GEOMETRY For Eercises 3 and 3, use the following information. The area of a rectangle is 84 square inches. Its length and width are both whole numbers. 3. What is the minimum perimeter of the rectangle? 38 in. 3. What is the maimum perimeter of the rectangle? 70 in. RENOVATION For Eercises 33 and 34, use the following information. Ms. Bater wants to tile a wall to serve as a splashguard above a basin in the basement. She plans to use equal-sized tiles to cover an area that measures 48 inches by 36 inches. 33. What is the maimum-size square tile Ms. Bater can use and not have to cut any of the tiles? -in. square 34. How many tiles of this size will she need? Reading to Learn Mathematics, p NAME DATE PERIOD Reading to Learn Mathematics Factors and Greatest Common Factors ELL Pre-Activity How are prime numbers related to the search for etraterrestrial life? Read the introduction to Lesson 9- at the top of page 474 in your tetbook. If each beep counts as one, what are the first two prime numbers? and 3 Reading the Lesson. Every whole number greater than is either composite or prime.. Complete each statement. a. In the prime factorization of a whole number, each factor is a prime number. b. In the prime factorization of a negative integer, all the factors are prime ecept the factor. Marching Bands Drum Corps International (DCI) is a nonprofit youth organization serving junior drum and bugle corps around the world. Members of these marching bands range from 4 to years of age. Source: Chapter 9 Factoring MARCHING BANDS For Eercises 63 and 64, use the following information. Central High s marching band has 75 members, and the band from Northeast High has 90 members. During the halftime show, the bands plan to march into the stadium from opposite ends using formations with the same number of rows. 63. If the bands want to match up in the center of the field, what is the maimum number of rows? How many band members will be in each row after the bands are combined? NUMBER THEORY For Eercises 65 and 66, use the following information. One way of generating prime numbers is to use the formula p, where p is a prime number. Primes found using this method are called Mersenne primes. For eample, when p, 3. The first Mersenne prime is Find the net two Mersenne primes. 7, Will this formula generate all possible prime numbers? Eplain your reasoning. No, it does not generate the first prime number,. Online Research Data Update What is the greatest known prime number? Visit to learn more. 3. Eplain why the monomial 5 y is not in factored form. The variable has an eponent that is greater than. 4. Eplain the steps used below to find the greatest common factor (GCF) of 84 and Write the prime factorization of Write the prime factorization of 0. Common prime factors:,, 3 Identify common prime factors of 84 and 0. 3 Multiply the common factors to find the GCF of 84 and 0. Helping You Remember 5. How can the two words that make up the term prime factorization help you remember what the term means? Sample answer: The word factorization reminds you that the number must be epressed as a product of factors. The word prime reminds you that with the possible eception of, all of the factors used must be prime numbers. 478 Chapter 9 Factoring NAME DATE PERIOD 9- Enrichment, p. 58 Finding the GCF by Euclid s Algorithm Finding the greatest common factor of two large numbers can take a long time using prime factorizations. This method can be avoided by using Euclid s Algorithm as shown in the following eample. Eample Find the GCF of 09 and 53. Divide the greater number, 53, by the lesser, Divide the remainder into the divisor above. Repeat this process until the remainder is zero. The last nonzero remainder is the GCF Answers 8. The factors of 96 whose sum when doubled is the least are and The factors of 96 whose sum when doubled is the greatest are and 96.

14 Finding the GCF of distances will help you make a scale model of the solar system. Visit webquest to continue work on your WebQuest project. 68a. false; countereample: a 3, b 4 SOL/EOC Practice Standardized Test Practice Maintain Your Skills Mied Review Getting Ready for the Net Lesson 84. y 4y 67. GEOMETRY The area of a triangle is 0 square centimeters. What are possible whole-number dimensions for the base and height of the triangle? See margin. 68. CRITICAL THINKING Suppose 6 is a factor of ab, where a and b are natural numbers. Make a valid argument to eplain why each assertion is true or provide a countereample to show that an assertion is false. a. 6 must be a factor of aorof b. b. 3 must be a factor of aorof b. True; see margin for eplanation. c. 3 must be a factor of aandof b. false; countereample: a 3, b WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See p. 5A. How are prime numbers related to the search for etraterrestrial life? Include the following in your answer: a list of the first 30 prime numbers and an eplanation of how you found them, and an eplanation of why a signal of this kind might indicate that an etraterrestrial message is to follow. 70. Miko claims that there are at least four ways to design a 0-square-foot rectangular space that can be tiled with -foot by -foot tiles. Which statement best describes this claim? D A Her claim is false because 0 is a prime number. B Her claim is false because 0 is not a perfect square. C Her claim is true because 40 is a multiple of 0. D Her claim is true because 0 has at least eight factors. 7. Suppose Ψ is defined as the largest prime factor of. For which of the following values of would Ψ have the greatest value? A A 53 B 74 C 99 D 7 Find each product. (Lessons 8-7 and 8-8) 73. 9a p 4 56p 6 7. ( ) (3a 5)(3a 5) 74. (7p 4)(7p 4) 75. (6r 7)(r 5) 76. (0h k)(h 5k) 77. (b 4)(b 3b 8) r 6r 35 0h 5hk 5k b 3 7b 6b 7 Find the value of r so that the line that passes through the given points has the given slope. (Lesson 5-) 78. (, ), (, r), m ( 5, 9), (r, 6), m RETAIL SALES A department store buys clothing at wholesale prices and then marks the clothing up 5% to sell at retail price to customers. If the retail price of a jacket is $79, what was the wholesale price? (Lesson 3-7) $63.0 PREREQUISITE SKILL Use the Distributive Property to rewrite each epression. (To review the Distributive Property, see Lesson -5.) 8. 5( 8) a(3a ) 3a a 83. g(3g 4) 6g 8g 84. 4y(3y 6) 85. 7b 7c 7(b c) ( 3) Lesson 9- Factors and Greatest Common Factors Assess Open-Ended Assessment Modeling Write a number and each step of its prime factorization (using the factor tree method) on note cards. Each factor, including the intermediate factors, should be on separate note cards. Then draw some arrows on other note cards. Tape the cards at random on the chalkboard. Have student volunteers come up and arrange the factors to find the prime factorization of the number. Getting Ready for Lesson 9- PREREQUISITE SKILL Students will learn to factor polynomials using the Distributive Property in Lesson 9-. Students should know how to rewrite epressions using the Distributive Property so they can factor polynomials. Use Eercises 8 86 to determine your students familiarity with the Distributive Property. Answers d d d d y y a a a b b g h 44. p q q n n n m y z z z a a b c c 67. base cm, height 40 cm; base cm, height 0 cm; base 4 cm, height 0 cm; base 5 cm, height 8 cm; base 8 cm, height 5 cm; base 0 cm, height 4 cm; base 0 cm, height cm; base 40 cm, height cm 68b.If 6 is a factor of ab, then the prime factorization of ab must contain 3 So, 3 must be a factor of either a or b. Lesson 9- Factors and Greatest Common Factors 479

15 Algebra Activity A Preview of Lesson 9- Getting Started Objective Factor polynomials with algebra tiles. Materials algebra tiles product mat Teach Remind students that they used rectangles at the beginning of Lesson 9- to find factors of whole numbers. The procedure for finding the factors of polynomials with algebra tiles is very similar. The length and width of a modeled polynomial represent the factors of the polynomial. Assess In Eercises 4, students need to recognize that they must arrange the tiles into a rectangle with a width greater than one in order to find the factors. In Eercises 5 9, students should recognize that the binomials that cannot be factored can only be modeled in a rectangle with a width of one. Study Notebook You may wish to have students summarize this activity and what they learned from it. Factoring Using the Distributive Property 480 Chapter 9 Factoring A Preview of Lesson 9- Sometimes you know the product of binomials and are asked to find the factors. This is called factoring. You can use algebra tiles and a product mat to factor binomials. Activity Use algebra tiles to factor 3 6. Model the polynomial 3 6. Activity Use algebra tiles to factor 4. Model the polynomial 4. Arrange the tiles into a rectangle. The total area of the rectangle represents the product, and its length and width represent the factors. The rectangle has a width of 3 and a length of. So, 3 6 3( ). 9. Binomials can be factored if they can be represented by a rectangle. Eamples: can be factored and cannot be factored. Model and Analyze Use algebra tiles to factor each binomial.. 0 ( 5). 6 8 (3 4) 3. 5 (5 ) (3 ) Tell whether each binomial can be factored. Justify your answer with a drawing yes no 7. yes 8. 3 no 5 8. See pp. 5A 5B for drawings. 9. MAKE A CONJECTURE Write a paragraph that eplains how you can use algebra tiles to determine whether a binomial can be factored. Include an eample of one binomial that can be factored and one that cannot. Resource Manager Virginia SOL STANDARD A. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. 3 Arrange the tiles into a rectangle. 4 The rectangle has a width of and a length of 4. So, 4 ( 4). Teaching Algebra with Manipulatives pp. 0 (master for algebra tiles) p. 7 (master for product mat) p. 56 (student recording sheet) Glencoe Mathematics Classroom Manipulative Kit algebra tiles product mat 480 Chapter 9 Factoring

16 Factoring Using the Distributive Property Lesson Notes Vocabulary factoring factoring by grouping Virginia SOL STANDARD A. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. Study Tip Look Back To review the Distributive Property, see Lesson -5. Factor polynomials by using the Distributive Property. Solve quadratic equations of the form a b 0. can you determine how long a baseball will remain in the air? Nolan Ryan, the greatest strike-out pitcher in the history of baseball, had a fastball clocked at 98 miles per hour or about 5 feet per second. If he threw a ball directly upward with the same velocity, the height h of the ball in feet above the point at which he released it could be modeled by the formula h 5t 6t, where t is the time in seconds. You can use factoring and the Zero Product Property to determine how long the ball would remain in the air before returning to his glove. FACTOR BY USING THE DISTRIBUTIVE PROPERTY In Chapter 8, you used the Distributive Property to multiply a polynomial by a monomial. a(6a 8) a(6a) a(8) a 6a You can reverse this process to epress a polynomial as the product of a monomial factor and a polynomial factor. a 6a a(6a) a(8) a(6a 8) Thus, a factored form of a 6a is a(6a 8). Factoring a polynomial means to find its completely factored form. The epression a(6a 8) is not completely factored since 6a 8 can be factored as (3a 4). Eample Use the Distributive Property Use the Distributive Property to factor each polynomial. a. a 6a First, find the GCF of a and 6a. a 3 a a Factor each number. 6a a Circle the common prime factors. GCF: a or 4a Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. a 6a 4a(3 a) 4a( ) Rewrite each term using the GCF. 4a(3a) 4a(4) Simplify remaining factors. 4a(3a 4) Distributive Property Thus, the completely factored form of a 6a is 4a(3a 4). Lesson 9- Factoring Using the Distributive Property 48 Focus 5-Minute Check Transparency 9- Use as a quiz or review of Lesson 9-. Mathematical Background notes are available for this lesson on p. 47C. Building on Prior Knowledge In Chapter, students were introduced to the Distributive Property and learned how to use it to simplify epressions. In Chapter 8, students learned to multiply a polynomial by a monomial using the Distributive Property. In this lesson, students will reverse that process to factor polynomials. can you determine how long a baseball will remain in the air? Ask students: What is the greatest common factor (GCF) of two numbers? The GCF is the greatest number that is a factor of both original numbers. What is the GCF of 5t and 6t? t What is the height of the ball when t 0? 0 ft What is the height of the ball when t? 35 ft Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 53 Practice, p. 53 Reading to Learn Mathematics, p. 533 Enrichment, p. 534 Assessment, p. 573 Parent and Student Study Guide Workbook, p. 69 Prerequisite Skills Workbook, pp. 3 4 School-to-Career Masters, p. 7 Transparencies 5-Minute Check Transparency 9- Answer Key Transparencies Technology Interactive Chalkboard Lesson - Lesson Title 48

17 Teach FACTOR BY USING THE DISTRIBUTIVE PROPERTY In-Class Eamples 3 Teaching Tip Tell students that one way to find the remaining factors is to divide each term by the GCF. Use the Distributive Property to factor each polynomial. a (3 5) b. y 4y 30 y 4 6y( 4y 5y 3 ) Power Point Teaching Tip Remind students that when using the FOIL method, they multiply the First terms, Outer terms, Inner terms, and Last terms. Factor y 7 y 7. ( )(y 7) Factor 5a 3ab 4b 0. ( 3a 4)(b 5) Concept Check Factoring Using the Distributive Property Malcolm and Fatima each factored the polynomial a 6c ab 3bc. Malcolm s answer was ( b)(a 3c) and Fatima s was (a 3c)( b). Which is correct? Eplain your answer. Both are correct. The order in which factors are multiplied does not affect the product. Study Tip Factoring by Grouping Sometimes you can group terms in more than one way when factoring a polynomial. For eample, the polynomial in Eample could have been factored in the following way. 4ab 8b 3a 6 (4ab 3a) (8b 6) a(4b 3) (4b 3) (4b 3)(a ) Notice that this result is the same as in Eample. TEACHING TIP Since the order in which factors are multiplied does not affect the product, ( 5 3)(y 7) is also a correct factoring of 35 5y 3y. b. 8cd c d 9cd 8cd 3 3 c d d c d 3 c c d 9cd 3 3 c d GCF: 3 c d or 3cd The Distributive Property can also be used to factor some polynomials having four or more terms. This method is called factoring by grouping because pairs of terms are grouped together and factored. The Distributive Property is then applied a second time to factor a common binomial factor. Eample Use Grouping Factor 4ab 8b 3a 6. 4ab 8b 3a 6 (4ab 8b) (3a 6) Group terms with common factors. 4b(a ) 3(a ) Factor the GCF from each grouping. (a )(4b 3) Distributive Property CHECK Factor each number. Circle the common prime factors. 8cd c d 9cd 3cd(6d) 3cd(4c) 3cd(3) 3cd(6d 4c 3) Use the FOIL method. F O I L (a )(4b 3) (a)(4b) (a)(3) ()(4b) ()(3) 4ab 3a 8b 6 Rewrite each term using the GCF. Distributive Property Recognizing binomials that are additive inverses is often helpful when factoring by grouping. For eample, 7 y and y 7 are additive inverses. By rewriting 7 y as (y 7), factoring by grouping is possible in the following eample. Eample 3 Use the Additive Inverse Property Factor 35 5y 3y. 35 5y 3y (35 5y) (3y ) Group terms with common factors. 5(7 y) 3(y 7) Factor the GCF from each grouping. 5( )(y 7) 3(y 7) 7 y (y 7) 5(y 7) 3(y 7) 5( ) 5 (y 7)( 5 3) Distributive Property Factoring by Grouping Words A polynomial can be factored by grouping if all of the following situations eist. There are four or more terms. Terms with common factors can be grouped together. The two common factors are identical or are additive inverses of each other. Symbols a b ay by (a b) y(a b) (a b)( y) 48 Chapter 9 Factoring 48 Chapter 9 Factoring

18 SOLVE EQUATIONS BY FACTORING factoring. Consider the following products. Some equations can be solved by 6(0) 0 0( 3) 0 (5 5)(0) 0 ( 3 3) 0 Notice that in each case, at least one of the factors is zero. These eamples illustrate the. Zero Product Property Zero Product Property Words If the product of two factors is 0, then at least one of the factors must be 0. Symbols For any real numbers a and b, if ab 0, then either a 0, b 0, or both a and b equal zero. Eample 4 Solve an Equation in Factored Form Solve (d 5)(3d 4) 0. Then check the solutions. If (d 5)(3d 4) 0, then according to the Zero Product Property either d 5 0 or 3d 4 0. (d 5)(3d 4) 0 Original equation d 5 0 or 3d 4 0 Set each factor equal to zero. d 5 3d 4 Solve each equation. The solution set is 5, 4 3. d 4 3 CHECK Substitute 5 and 4 for d in the original equation. 3 (d 5)(3d 4) 0 (d 5)(3d 4) 0 (5 5)[3(5) 4] (0)(9) 0 9 (0) SOLVE EQUATIONS BY FACTORING In-Class Eamples 4 5 Power Point Teaching Tip If students think using the Zero Product Property to set factors equal to zero is somewhat arbitrary, have them multiply the two factors to obtain a polynomial, and set the polynomial equal to zero. Then have students substitute the two solutions into the polynomial to see that they produce a true sentence. Solve ( )(4 ) 0. Then check the solutions., 4 Teaching Tip Remind students that for the Zero Product Property to work, one of two factors is equal to zero. Therefore, students must factor 7 before they can assume that one term is equal to zero. Solve 4y y. Then check the solutions. 0, 3 If an equation can be written in the form ab 0, then the Zero Product Property can be applied to solve that equation. Study Tip Common Misconception You may be tempted to try to solve the equation in Eample 5 by dividing each side of the equation by. Remember, however, that is an unknown quantity. If you divide by, you may actually be dividing by zero, which is undefined. Eample 5 Solve an Equation by Factoring Solve 7. Then check the solutions. Write the equation so that it is of the form ab 0. 7 Original equation 7 0 Subtract 7 from each side. ( 7) 0 Factor the GCF of and 7, which is. 0 or 7 0 Zero Product Property 7 Solve each equation. The solution set is {0, 7}. Check by substituting 0 and 7 for in the original equation. Lesson 9- Factoring Using the Distributive Property 483 Differentiated Instruction Visual/Spatial When students solve factored equations, have them write each factor on a separate sheet of scrap paper, followed by 0 on a third sheet of scrap paper. Then, have students remove one of the factors and solve the remaining equation. Once the first solution is found, have students place the other factor equal to zero and solve for the second solution. Lesson 9- Factoring Using the Distributive Property 483

19 3 Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 9. include eplanations on how to factor polynomials using the distributive property, and how to solve equations using factoring. include any other item(s) that they find helpful in mastering the skills in this lesson. Concept Check. an equation that can be written as a product of factors that equal 0 Guided Practice GUIDED PRACTICE KEY Eercises Eamples , 5. Write 4 as a product of factors in three different ways. Then decide which of the three is the completely factored form. Eplain your reasoning. See margin.. OPEN ENDED Give an eample of the type of equation that can be solved by using the Zero Product Property. 3. Eplain why ( )( 4) 0 cannot be solved by dividing each side by. The division would eliminate as a solution. Factor each polynomial. 7. ab(ab 4 8ab) ( 4) 5. 6z 40z 8z( 5z) 6. 4mnp 36mnp mnp(p 3n) 7. a3b 8ab 6ab3 8. 5y 5y 4y (5y 4)(y 3) 9. 5c 0c d 4cd (5c d)( c) Solve each equation. Check your solutions. 0. h(h 5) 0 {0, 5} Application. (n 4)(n ) 0 {, 4} 冦 5. 5m 3m 0, 3 PHYSICAL SCIENCE For Eercises 3 5, use the information below and in the graphic. A flare is launched from a life raft. The height h of the flare in feet above the sea is modeled by the formula h 00t 6t, where t is the time in seconds after the flare is launched. 3. At what height is the flare when it returns to the sea? 0 ft h 00t 6t 4. Let h 0 in the equation h 00t 6t and solve for t. 0, How many seconds will it take for the flare to return to the sea? Eplain your reasoning. About the Eercises Organization by Objective Factor by Using the Distributive Property: 6 39 Solve Equations by Factoring: Odd/Even Assignments Eercises 6 39 and are structured so that students practice the same concepts whether they are assigned odd or even problems. 00 ft /s 6.5 s; The answer 0 is not reasonable since it represents the time at which the flare is launched. h 0 indicates increased difficulty Practice and Apply Factor each polynomial See p. 5A. For Eercises See Eamples 6 9, , 3 4, 5 Etra Practice See page 840. Assignment Guide Basic: 7 39 odd, 40 43, odd, 6 8 Average: 7 39 odd, 4, 43, 45, 47 6 odd, 6 8 Advanced: 6 38 even, 44, 45, even, 6 75 (optional: 76 8) All: Practice Quiz ( 0) y 7. 6a 4b 8. a5b a 9. 3y 0. cd 3d. 4gh 8h. 5ay 30ay 3. 8bc 4bc 4. yz 40y3z 6. a 7. 5y 5y 5. 8abc 8. a abc3 0b 3c ab 9. 3p3q 3. 4 a3b3 9pq 36pq y 9y 8y a 5a 8a a 3ay 4b 3by 37. my 7 7m y 38. 8a 6 a y 5 y GEOMETRY For Eercises 40 and 4, use the following information. A quadrilateral has 4 sides and diagonals. A pentagon has 5 sides and 5 diagonals. 3 You can use n n to find the number of diagonals in a polygon with n sides. 40. Write this epression in factored form. n(n 3) 4. Find the number of diagonals in a decagon (0-sided polygon) Chapter 9 Factoring Answers. Sample answers: 4( 3), (4 ), or 4( 3); 4( 3); 4 is the GCF of 4 and. 484 Chapter 9 Factoring 冧 63. Answers should include the following. Let h 0 in the equation h 5t 6t. To solve 0 5t 6t, factor the right-hand side as t(5 6t). Then, since t(5 6t) 0, either t 0 or 5 6t 0. Solving each equation for t, we find that t 0 or t The solution t 0 represents the point at which the ball was initially thrown into the air. The solution t 9.44 represents how long it took after the ball was thrown for it to return to the same height at which it was thrown.

20 Marine Biologist Marine biologists study factors that affect organisms living in and near the ocean. Online Research For information about a career as a marine biologist, visit: careers Source: National Sea Grant Library SOFTBALL For Eercises 4 and 43, use the following information. Albertina is scheduling the games for a softball league. To find the number of games she needs to schedule, she uses the equation g n n, where g represents the number of games needed for each team to play each other team eactly once and n represents the number of teams. 4. Write this equation in factored form. g n(n ) 43. How many games are needed for 7 teams to play each other eactly 3 times? 63 games GEOMETRY Write an epression in factored form for the area of each shaded region r r b a 4(a b 4) r (4 ) GEOMETRY Find an epression for the area of a square with the given perimeter. 46. P 0y in. 47. P 36a 6b cm 9 30y 5y in 8a 7ab 6b cm Solve each equation. Check your solutions. 48. ( 4) 0 {0, 4} 49. a(a 6) 0 { 6, 0} 50. (q 4)(3q 5) 0 { 4, 5} 5. (3y 9)(y 7) 0 { 3, 7} 5. (b 3)(3b 8) 0 3, (4n 5)(3n 7) 0 5 4, z z 0 { 4, 0} 55. 7d 35d 0 {0, 5} , , , , MARINE BIOLOGY In a pool at a water park, a dolphin jumps out of the water traveling at 0 feet per second. Its height h, in feet, above the water after t seconds is given by the formula h 0t 6t. How long is the dolphin in the air before returning to the water?.5 s 6. BASEBALL Malik popped a ball straight up with an initial upward velocity of 45 feet per second. The height h, in feet, of the ball above the ground is modeled by the equation h 45t 6t. How long was the ball in the air if the catcher catches the ball when it is feet above the ground? about.8 s 6. CRITICAL THINKING Factor a y a b y a y b b y. (a b )(a y b y ) 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can you determine how long a baseball will remain in the air? Include the following in your answer: an eplanation of how to use factoring and the Zero Product Property to find how long the ball would be in the air, and an interpretation of each solution in the contet of the problem. Lesson 9- Factoring Using the Distributive Property 485 Study Study Guide and and Intervention Intervention, p. 59 (shown) and p NAME DATE PERIOD Factoring Using the Distributive Property Factor by Using the Distributive Property The Distributive Property has been used to multiply a polynomial by a monomial. It can also be used to epress a polynomial in factored form. Compare the two columns in the table below. Multiplying Factoring 3(a b) 3a 3b 3a 3b 3(a b) (y z) y z y z (y z) 6y( ) 6y() 6y() y 6y 6y() 6y() y 6y 6y( ) Eample Use the Distributive Eample Factor Property to factor mn 80m. 6a 3ay b by by grouping. Find the GCF of mn and 80m. 6a 3ay b by mn 3 m n (6a 3ay) (b by) 80m 5 m m 3a( y) b( y) (3a b)( y) GCF m or 4m Check using the FOIL method. Write each term as the product of the GCF and its remaining factors. (3a b)( y) mn 80m 3a() (3a)( y) (b)() (b)( y) 4m(3 n) 4m( 5 m) 6a 3ay b by 4m(3n) 4m(0m) 4m(3n 0m) Thus mn 80m 4m(3n 0m). Eercises Factor each polynomial y. 30mn m n 6n 3. q 4 8q 3 q 4( y) n(30mn m 6) q(q 3 8q ) m 6n 8mn 6. 45s 3 5s 3(3 ) (m 3n 4mn) 5s (3s ) 7. 4c 3 4c 5 49c p p 4 44p y 3 8y y 7c 3 ( 6c 7c) p (5 p 4p 3 ) y(4y 8y ) a b 8ab 7ab. 6y 8z 4( 3 4 ) ab(4a 8b 7) (3y 6 4z) y 4y 3y 5. 4m 4mn 3mn 3n ( )( ) (y )(3y ) (4m 3n)(m n) 6. a 3z 4ay yz 7. a 3a 8a 8. a ya y (3 y)(4a z) (4a )(3a ) ( y)(a ) Skills Practice, p. 53 and NAME DATE PERIOD 9- Practice (Average) Practice, Factoring Using p. 53 the Distributive (shown) Property Factor each polynomial ab. 4d r s 3rs 8(8 5ab) 4(d 4) 3rs(r s) 4. 5cd 30c d 5. 3a 4b 6. 36y 48 y 5cd( cd) 8(4a 3b ) y(3y 4) y 35 y 8. 9c 3 d 6cd b c 3 60bc 3 5 y(6 7y) 3cd (3c d) 5bc 3 (5b 4) 0. 8p q 4pq 3 6pq. 5 3 y 0 y 5. 9a 3 8b 4c 8pq(pq 3q ) 5( y y 5) 3(3a 6b 8c) a 3a 6a b b b 6 ( )( 4) (a 3)(a 3) (4b )(b 3) 6. 6y 8 5y mn 4m 8n 8. a 5ab 6a 0b ( 5)(3y 4) ( m 6)(3n ) (3a 4)(4a 5b) Solve each equation. Check your solutions. 9. ( 3) b(b 4) 0. ( y 3)( y ) 0 {0, 3} { 4, 0} {, 3}. (a 6)(3a 7) 0 3. (y 5)( y 4) 0 4. (4y 8)(3y 4) 0 6, 7 5,4, 4 5. z 0z p 4p { 0, 0} 0, {0, 3} , 3,0 3,0 LANDSCAPING For Eercises 3 and 3, use the following information. A landscaping company has been commissioned to design a triangular flower bed for a mall entrance. The final dimensions of the flower bed have not been determined, but the company knows that the height will be two feet less than the base. The area of the flower bed can be represented by the equation A b b. 3. Write this equation in factored form. A b b 3. Suppose the base of the flower bed is 6 feet. What will be its area? ft 33. PHYSICAL SCIENCE Mr. Alim s science class launched a toy rocket from ground level with an initial upward velocity of 60 feet per second. The height h of the rocket in feet above the ground after t seconds is modeled by the equation h 60t 6t.How long was the rocket in the air before it returned to the ground? 3.75 s Reading to Learn Mathematics, p NAME DATE PERIOD Reading to Learn Mathematics Factoring Using the Distributive Property ELL Pre-Activity How can you determine how long a baseball will remain in the air? Read the introduction to Lesson 9- at the top of page 48 in your tetbook. In the formula h 5t 6t,what does the number 5 represent? the velocity of the baseball in feet per second at the instant it is thrown Reading the Lesson. Factoring a polynomial means to find its completely factored form. a. The epression (6 9) is a factored form of the polynomial 6 9.Why is this not its completely factored form? The numbers 6 and 9 have a common factor of 3, which needs to be factored out of the epression. b. Provide an eample of a completely factored polynomial. Sample answer: b(5b 4) is the completely factored form of 0b 8b. Lesson 9- Lesson 9- NAME DATE PERIOD 9- Enrichment, p. 534 c. Provide an eample of a polynomial that is not completely factored. Sample answer: b(0b 8) is a factored form of 0b 8b, but is not the completely factored form. Perfect, Ecessive, Defective, and Amicable Numbers A perfect number is the sum of all of its factors ecept itself. Here is an eample: There are very few perfect numbers. Most numbers are either ecessive or defective. An ecessive number is greater than the sum of all of its factors ecept itself. A defective number is less than this sum. Two numbers are amicable if the sum of the factors of the first number, ecept for the number itself, equals the second number, and vice versa. Solve each problem.. Write the perfect numbers between 0 and 3. 6, 8. The polynomial 5ab 5b 3a 6b can be rewritten as 5b(a b) 3(a b). Does this indicate that the original polynomial can be factored by grouping? Eplain. No, because 5b(a b) and 3(a b) do not have a common factor other than. 3. The polynomial 3 3y y can be rewritten as 3( y) ( y). Does this indicate that the original polynomial can be factored by grouping? Eplain. Yes, because 3( y) and ( y) have the common factor ( y). Helping You Remember 4. How would you eplain to a classmate when it is possible to use the Zero Product Property to solve an equation? The equation must have 0 on one side, and the other side must be the product of two epressions. Lesson 9- Factoring Using the Distributive Property 485

21 4 Assess Open-Ended Assessment Speaking Ask a volunteer to describe the similarities and differences between factoring using grouping, and factoring using the additive inverse property. Encourage other students to ask questions. Getting Ready for Lesson 9-3 PREREQUISITE SKILL Students will learn to factor trinomials in Lesson 9-3. It is important that students recall how to multiply polynomials to check that they correctly factored trinomials. Use Eercises 76 8 to determine your students familiarity with multiplying polynomials. Assessment Options Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 9- and 9-. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not yet mastered. Quiz (Lessons 9- and 9-) is available on p. 573 of the Chapter 9 Resource Masters. Answer., 3, 5, 9, 5, 5, 45, 75, 5; composite Maintain Your Skills P Standardized Test Practice SOL/EOC Practice Mied Review 66., 3, 4, 3; composite 67.,, 3, 4, 5, 6, 0,, 5, 0, 5, 30, 50, 60, 75, 00, 50, 300; composite 7. 9k 48k 64 Getting Ready for the Net Lesson 76. n n b 3b 70 ractice Quiz. Find the factors of 5. Then classify the number as prime or composite. (Lesson 9-) See margin.. Find the prime factorization of 30. (Lesson 9-) Factor 78a bc 3 completely. (Lesson 9-) 3 3 a a b c c c 4. Find the GCF of 54 3, 4 y, and 30y. (Lesson 9-) 6 Factor each polynomial. (Lesson 9-) 64. The total number of feet in yards, y feet, and z inches is A z A 3 y. B ( y z). y C 3y 36z. D z Lola is batting for her school s softball team. She hit the ball straight up with an initial upward velocity of 47 feet per second. The height h of the softball in feet above ground after t seconds can be modeled by the equation h 6t 47t 3. How long was the softball in the air before it hit the ground? C A 0.06 s B.5 s C 3 s D 3.5 s Find the factors of each number. Then classify each number as prime or composite. (Lesson 9-) , 67; prime Find each product. (Lesson 8-8) 69. 6s 6 4s p 5q 69. (4s 3 3) 70. (p 5q)(p 5q) 7. (3k 8)(3k 8) Simplify. Assume that no denominator is equal to zero. (Lesson 8-) 4 7. s s 7 s y 3 34p7qr 5 y4 y ( p3 qr ) p r FINANCE Michael uses at most 60% of his annual FlynnCo stock dividend to purchase more shares of FlynnCo stock. If his dividend last year was $885 and FlynnCo stock is selling for $4 per share, what is the greatest number of shares that he can purchase? (Lesson 6-) 37 shares PREREQUISITE SKILL Find each product. (To review multiplying polynomials, see Lesson 8-7.) 76. (n 8)(n 3) 77. ( 4)( 5) 78. (b 0)(b 7) 79. (3a )(6a 4) 80. (5p )(9p 3) 8. (y 5)(4y 3) 8a 6a 4 45p 33p 6 8y 4y y y y(4y ) 6. 3a b 40b 3 8a b 7. 6py 6p 5y 40 8b(4a 5b a b) (p 5)(3y 8) Solve each equation. Check your solutions. (Lesson 9-), 0 8. (8n 5)(n 4) 0 5 8, {0, 3} Lessons 9- and Chapter 9 Factoring 486 Chapter 9 Factoring

22 A Preview of Lesson 9-3 Virginia SOL STANDARD A. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. Factoring Trinomials You can use algebra tiles to factor trinomials. If a polynomial represents the area of a rectangle formed by algebra tiles, then the rectangle s length and width are factors of the area. Activity Use algebra tiles to factor 6 5. Model the polynomial 6 5. Place the tile at the corner of the product mat. Arrange the tiles into a rectangular array. Because 5 is prime, the 5 tiles can be arranged in a rectangle in one way, a -by-5 rectangle. Complete the rectangle with the tiles. The rectangle has a width of and a length of 5. Therefore, ( )( 5). Activity Use algebra tiles to factor 7 6. Model the polynomial Place the tile at the corner of the product mat. Arrange the tiles into a rectangular array. Since 6 3, try a -by-3 rectangle. Try to complete the rectangle. Notice that there are two etra tiles. Algebra Activity A Preview of Lesson 9-3 Getting Started Objective Factor trinomials with algebra tiles. Materials algebra tiles product mat Teach Remind students that they used algebra tiles in an activity prior to Lesson 9- to factor polynomials. Ask students to tell you what they must be able to form with the tiles in order to factor a polynomial. a rectangle Activity Remind students to read the width of the tiles along the edge of the rectangle. The tile has a width of and the tiles have a width of one. Activity Encourage students to try several different arrangements until they can form a rectangle. While the tile should be in the corner, there is more than one correct way to arrange the tiles into a rectangle. (continued on the net page) Algebra Activity Factoring Trinomials 487 Resource Manager Teaching Algebra with Manipulatives pp. 0 (master for algebra tiles) p. 7 (master for product mat) p. 59 (student recording sheet) Glencoe Mathematics Classroom Manipulative Kit algebra tiles product mat Algebra Activity Factoring Trinomials 487

23 Algebra Activity Activity 3 Remind students to pay close attention to the sign of the tiles. It would be very easy to mistake the factors as ( 3)( 3) if they do not pay attention to the signs. Students may need to be reminded that adding a zero pair is similar to adding the same number to both sides of an equation. Be sure that students are careful to add one tile and one tile when they add a zero pair. Assess After students complete Eercises 8, ask them whether they notice a correlation between the need to use zero pairs to factor the trinomial, and the appearance of the resulting factors. Sample answer: When zero pairs are used, the signs of the factors are opposite. When zero pairs are not used, the signs of the factors are the same. Study Notebook You may wish to have students summarize this activity and what they learned from it. Arrange the tiles into a -by-6 rectangular array. This time you can complete the rectangle with the tiles. The rectangle has a width of and a length of 6. Therefore, 7 6 ( )( 6). Activity 3 Use algebra tiles to factor 3. Model the polynomial 3. Place the tile at the corner of the product mat. Arrange the tiles into a -by-3 rectangular array as shown. 6 Place the tile as shown. Recall that you can add zero-pairs without changing the value of the polynomial. In this case, add a zero pair of tiles. The rectangle has a width of and a length of 3. Therefore, 3 ( )( 3). 3 zero pair Model. ( 3)( ). ( 4)( ) 3. ( 3)( ) 4. ( )( ) Use algebra tiles to factor each trinomial ( 4)( 3) ( )( ) ( )( ) ( 4)( ) 488 Chapter 9 Factoring 488 Chapter 9 Factoring

24 Study Tip Reading Math A quadratic trinomial is a trinomial of degree. This means that the greatest eponent of the variable is. Factoring Trinomials: b c Factor trinomials of the form b c. Solve equations of the form b c 0. can factoring be used to find the dimensions of a garden? Tamika has enough bricks to make a 30-foot border around the rectangular vegetable garden she is planting. The booklet she got from the nursery says that the plants will need a space of 54 square feet to grow. What should the dimensions of her garden be? To solve this problem, you need to find two numbers whose product is 54 and whose sum is 5, half the perimeter of the garden. Words A 54 ft P 30 ft FACTOR b c In Lesson 9-, you learned that when two numbers are multiplied, each number is a factor of the product. Similarly, when two binomials are multiplied, each binomial is a factor of the product. To factor some trinomials, you will use the pattern for multiplying two binomials. Study the following eample. F O I L ( )( 3) ( ) ( 3) ( ) ( 3) Use the FOIL method. 3 6 Simplify. (3 ) 6 Distributive Property 5 6 Simplify. Observe the following pattern in this multiplication. ( )( 3) (3 ) ( 3) ( m)( n) (n m) mn (m n) mn Virginia SOL STANDARD A. The student will factor completely firstand second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. b c b m n and c mn Notice that the coefficient of the middle term is the sum of m and n and the last term is the product of m and n. This pattern can be used to factor quadratic trinomials of the form b c. Factoring b c To factor quadratic trinomials of the form b c, find two integers, m and n, whose sum is equal to b and whose product is equal to c. Then write b c using the pattern ( m)( n). Symbols b c ( m)( n) when m n b and mn c. Eample 5 6 ( )( 3), since 3 5 and 3 6. Lesson Notes Focus 5-Minute Check Transparency 9-3 Use as a quiz or review of Lesson 9-. Mathematical Background notes are available for this lesson on p. 47D. can factoring be used to find the dimensions of a garden? Ask students: Why do you need to find two numbers whose product is 54 to find the dimensions of the garden? The garden is a rectangle with an area of 54 ft. The area of a rectangle is equal to the length times the width. Since the area is 54 ft, the length and the width must be two numbers whose product is 54. What two integers have a product of 54? and 54, and 7, 3 and 8, 6 and 9 Which pair has a sum of 5? 6 and 9 What are the dimensions of the vegetable garden? The garden is 6 ft by 9 ft. Lesson 9-3 Factoring Trinomials: b c 489 Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 537 Practice, p. 538 Reading to Learn Mathematics, p. 539 Enrichment, p. 540 Assessment, pp. 573, 575 Parent and Student Study Guide Workbook, p. 70 Transparencies 5-Minute Check Transparency 9-3 Answer Key Transparencies Technology AlgePASS: Tutorial Plus, Lessons 4, 5 Interactive Chalkboard Lesson - Lesson Title 489

25 To determine m and n, find the factors of c and use a guess-and-check strategy to find which pair of factors has a sum of b. Teach Eample b and c Are Positive FACTOR b c Factor 6 8. In this trinomial, b 6 and c 8. You need to find two numbers whose sum is 6 and whose product is 8. Make an organized list of the factors of 8, and look for the pair of factors whose sum is 6. The concept of factoring trinomials as introduced in this lesson may seem somewhat abstract to some students. Whenever you introduce abstract concepts, it is good to reinforce them with a concrete eample. After introducing factoring trinomials, refer students back to the lesson opener problem. Ask students to describe any similarities they notice between finding the dimensions of the garden and factoring a trinomial. New Factors of 8 Sum of Factors, 8, The correct factors are and ( m)( n) Write the pattern. ( )( 4) m and n 4 CHECK You can check this result by multiplying the two factors. F O I L ( )( 4) 4 8 FOIL method 6 8 Simplify. When factoring a trinomial where b is negative and c is positive, you can use what you know about the product of binomials to help narrow the list of possible factors. Eample b Is Negative and c Is Positive In-Class Eamples Factor 0 6. Power Point Teaching Tip Tell students that the order in which they record the factors does not matter. So, ( 4)( ) is also correct. Factor 7. ( 3)( 4) In this trinomial, b 0 and c 6. This means that m n is negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 6, and look for the pair of factors whose sum is 0. Study Tip Testing Factors Once you find the correct factors, there is no need to test any other factors. Therefore, it is not necessary to test 4 and 4 in Eample. Teaching Tip If students use the graphing calculator to check their factoring, make sure they clear all other functions from the Y= list, and clear all other drawings from the draw menu. Factor 7. ( 3)( 9) TEACHING TIP Caution students that two graphs may appear to coincide in the standard viewing window, but they do not. Have them use the TABLE feature to verify the identical y values. Factors of 6 Sum of Factors, 6, 8 4, The correct factors are and ( m)( n) Write the pattern. ( )( 8) m and n 8 CHECK You can check this result by using a graphing calculator. Graph y 0 6 and y ( )( 8) on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly. [ 0, 0] scl: by [ 0, 0] scl: You will find that keeping an organized list of the factors you have tested is particularly important when factoring a trinomial like, where the value of c is negative. 490 Chapter 9 Factoring Differentiated Instruction Kinesthetic As students are learning the rules for factoring trinomials, encourage them to use algebra tiles to confirm their results. Students should soon realize that the greater the values of b and c in the trinomials, the more cumbersome algebra tiles become, which should reinforce the importance of learning to factor using the method in the tet. 490 Chapter 9 Factoring

26 Study Tip Alternate Method You can use the opposite of FOIL to factor trinomials. For instance, consider Eample 3. ( )( ) Try factor pairs of until the sum of the products of the Inner and Outer terms is. Eample 3 Factor. b Is Positive and c Is Negative In this trinomial, b and c. This means that m n is positive and mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of, where one factor of each pair is negative. Look for the pair of factors whose sum is. Factors of Sum of Factors,,, 6 4, 6 4 3, 4 3, 4 The correct factors are 3 and 4. ( m)( n) Write the pattern. ( 3)( 4) m 3 and n 4 Eample 4 b Is Negative and c Is Negative Factor 7 8. Since b 7 and c 8, m n is negative and mn is negative. So either m or n is negative, but not both. Factors of 8 Sum of Factors, 8 7, 8 7, 9 7 The correct factors are and ( m)( n) ( )( 9) Write the pattern. m and n 9 SOLVE EQUATIONS BY FACTORING Some equations of the form b c 0 can be solved by factoring and then using the Zero Product Property. Eample 5 Solve an Equation by Factoring Solve 5 6. Check your solutions. 5 6 Original equation Rewrite the equation so that one side equals 0. ( )( 6) 0 Factor. 0 or 6 0 Zero Product Property 6 Solve each equation. The solution set is {, 6}. CHECK Substitute and 6 for in the original equation () 5() 6 ( 6) 5( 6) In-Class Eamples Teaching Tip Point out to students that whenever c is negative, either m or n must be negative, because if a product is negative, then one of the two factors must be negative. Factor 3 8. ( 6)( 3) Factor 0. ( 5)( 4) SOLVE EQUATIONS BY FACTORING In-Class Eamples Teaching Tip Tell students to be careful when rewriting equations so that one side equals zero. They must perform the same operation on both sides of the equation, and they must pay attention to the signs in the resulting equation. Solve 5. Check your solutions. { 5, 3} ARCHITECTURE Marion has a small art studio in her backyard. She wants to build a new studio that has three times the area of the old studio by increasing the length and width by the same amount. What are the dimensions of the new studio? ft Eisting Studio 0 ft Power Point Power Point Lesson 9-3 Factoring Trinomials: b c 49 The dimensions of the new studio should be 8 ft by 0 ft. Lesson 9-3 Factoring Trinomials: b c 49

27 3 Practice/Apply Study Notebook Have students include eplanations on how to factor trinomials. include any other item(s) that they find helpful in mastering the skills in this lesson. FIND THE ERROR Ask students to recall how many answers they usually get when solving a trinomial equation. About the Eercises Organization by Objective Factor b c: 7 34 Solve Equations by Factoring: Odd/Even Assignments Eercises 7 53 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Eercises require a graphing calculator. Assignment Guide Basic: 7 33 odd, 37 5 odd, 55, 57 60, 63 65, Average: 7 55 odd, 57 60, 63 65, (optional: 66 69) Advanced: 8 56 even, (optional: 78 83) Concept Check. Sample answer: ; {4, 0} GUIDED PRACTICE KEY Eercises Eamples Guided Practice 49 Chapter 9 Factoring Eample 6 Solve a Real-World Problem by Factoring YEARBOOK DESIGN Asponsor for the school yearbook has asked that the length and width of a photo in their ad be increased by the same amount in order to double the area of the photo. If the photo was originally centimeters wide by 8 centimeters long, what should the new dimensions of the enlarged photo be? Eplore Plan Begin by making a diagram like the one shown above, labeling the appropriate dimensions. Let the amount added to each dimension of the photo. The new length times the new width equals the new area. 8 (8)() Solve ( )( 8) (8)() Write the equation. Eamine Multiply. ( 4)( 4) 0 Factor. Subtract 9 from each side. 4 0 or 4 0 Zero Product Property 4 4 Solve each equation. old area The solution set is { 4, 4}. Only 4 is a valid solution, since dimensions cannot be negative. Thus, the new length of the photo should be 4 or 6 centimeters, and the new width should be 4 8 or centimeters.. Eplain why, when factoring 6 9, it is not necessary to check the sum of the factor pairs and 9 or 3 and 3. See margin.. OPEN ENDED Give an eample of an equation that can be solved using the factoring techniques presented in this lesson. Then, solve your equation. 3. FIND THE ERROR Peter and Aleta are solving 5. Peter + = 5 ( + ) = 5 = 5 or + = 5 = 3 Who is correct? Eplain your reasoning. Aleta; to use the Zero Product Property, one side of the equation must equal zero. Factor each trinomial. 4. ( 3)( 8) 5. (c )(c ) 6. (n 3)(n 6) c 3c 6. n 3n p p a a 9. 4y 3y (p 5)(p 7) (a 3)(a 4) ( 3y)( y) Aleta + = = 0 ( - 3)( + 5) = 0-3 = 0 or + 5 = 0 = 3 = -5 8 Answers. In this trinomial, b 6 and c 9. This means that m n is positive and mn is positive. Only two positive numbers have both a positive sum and product. Therefore, negative factors of 9 need not be considered. 0. {, 6}. { 9, 4}. {, } 3. { 9, } 4. {, } 5. { 7, 0} 7. (a 3)(a 5) 8. ( 3)( 9) 9. (c 5)(c 7) 0. (y 0)(y 3). (m )(m ). (d 5)(d ) 3. (p 8)(p 9) 4. (g 4)(g 5) 5. ( )( 7) 6. (b 4)(b 5) 7. (h 5)(h 8) 8. (n 6)(n 9) 9. (y 7)(y 6) 30. (z )(z 0) 3. (w )(w 6) 3. ( )( 5) 33. (a b)(a 4b) 34. ( 4y)( 9y) 49 Chapter 9 Factoring

28 Solve each equation. Check your solutions See margin. 0. n 7n 6 0. a 5a p 9p y 9 0y d 3d 70 Study Study Guide and and Intervention Intervention, p. 535 (shown) and p NAME DATE PERIOD Factoring Trinomials: b c Factor b c To factor a trinomial of the form b c, find two integers, m and n, whose sum is equal to b and whose product is equal to c. Application 6. NUMBER THEORY Find two consecutive integers whose product is 56. and 3 or 3 and indicates increased difficulty Practice and Apply For See Eercises Eamples , 6 6, 6 Etra Practice See page { 4, } 38. { 8, } 43. {3, 6} 44. {4, 8} 46. { 5, } Factor each trinomial See margin. 7. a 8a c c y 3y 30. m m. d 7d 0 3. p 7p 7 4. g 9g b b 0 7. h 3h n 3n y y z 8z w w a 5ab 4b 34. 3y 36y GEOMETRY Find an epression for the perimeter of a rectangle with the given area. 35. area area Solve each equation. Check your solutions b 0b y 4y 0 { 6, } 40. d d 8 0 { 4, } 4. a 3a 8 0 { 4, 7} 4. g 4g 45 0 { 5, 9} 43. m 9m n n z 8 7z {, 9} 46. h 5 6h k 0k {4, 6} { 4, 5} 49. c 50 3c { 5, }50. y 9y 54 {, 7} 5. 4p p 5 { 7, 3} { 8, 0} {4, 4} 54. SUPREME COURT When the Justices of the Supreme Court assemble to go on the Bench each day, each Justice shakes hands with each of the other Justices for a total of 36 handshakes. The total number of handshakes h possible for n people is given by h n n. Write and solve an equation to determine the number of Justices on the Supreme Court. 36 n n ; NUMBER THEORY Find two consecutive even integers whose product is and or and GEOMETRY The triangle has an area of 40 square centimeters. Find the height h of the triangle. 5 cm h cm Factoring b c b c ( m)( n), where m n b and mn c. Eample Factor each trinomial. Eample a. 7 0 In this trinomial, b 7 and c 0. Factors of 0 Since 5 7 and 5 0, let m and n ( 5)( ) b. 8 7 Sum of Factors, 0, 5 7 In this trinomial, b 8 and c 7. Notice that m n is negative and mn is positive, so m and n are both negative. Since 7 ( ) 8 and ( 7)( ) 7, m 7 and n. 8 7 ( 7)( ) Eercises Factor each trinomial. Factor 6 6. In this trinomial, b 6 and c 6. This means m n is positive and mn is negative. Make a list of the factors of 6, where one factor of each pair is positive. Factors of 6 Sum of Factors, 6 5, 6 5, 8 6, 8 6 Therefore, m and n ( )( 8) m m 3 3. r 3r ( 3)( ) (m 4)(m 8) (r )(r ) ( 3)( ) ( 7)( 3) ( )( ) 7. c 4c 8. p 6p (c )(c 6) (p 8)(p 8) (9 )( ) a 8a 9. y 7y 8 ( 5)( ) (a )(a 9) (y 8)( y ) y 4y 3 5. m 9m 0 ( 3)( ) (y )(y 3) (m 4)(m 5) a 4a y y ( 0)( ) (a )(a ) (9 y)( y) 9. y y 0. a 4ab 4b. 6y 7y ( y)( y) (a b)(a b) ( 7y)( y) NAME DATE PERIOD 9-3 Skills Practice Practice, p. 537 and (Average) Practice, Factoring Trinomials: p. 538 (shown) b c Factor each trinomial.. a 0a 4. h h (a 4)(a 6) (h 3)(h 9) ( )( 3) 4. g g w w y 4y 60 (g 7)(g 9) (w 8)(w 7) (y 0)(y 6) 7. b 4b 3 8. n 3n 8 9. c 4c 45 (b 4)(b 8) (n 7)(n 4) (c 5)(c 9) 0. z z 30. d 6d (z 6)(z 5) (d 9)(d 7) ( 3)( 8) 3. q q r r (q 8)(q 7) ( 5)( ) (r 6)(r ) g g 7. j 9jk 0k 8. m mv 56v (g )(g 4) (j 0k)(j k) (m 8v)(m 7v) Solve each equation. Check your solutions p 5p k 3k 54 0 { 4, 3} {, 7} { 9, 6}. b b n 4n 3 4. h 7h 60 { 4, 6} { 8, 4} {5, } 5. c 6c z 4z 7 7. y 84 5y {, 8} { 4, 8} { 7, } a 8a 9. u 6u s s 5 {8, 0} {, 8} { 3, 4) 3. Find all values of k so that the trinomial k 35 can be factored using integers. 34,,, 34 CONSTRUCTION For Eercises 3 and 33, use the following information. A construction company is planning to pour concrete for a driveway. The length of the driveway is 6 feet longer than its width w. 3. Write an epression for the area of the driveway. w (w 6) ft 33. Find the dimensions of the driveway if it has an area of 60 square feet. 0 ft by 6 ft Lesson 9-3 Supreme Court The Conference handshake has been a tradition since the late 9th century. Source: CRITICAL THINKING Find all values of k so that each trinomial can be factored using integers. 57. k 9 8, k 4 5, 9, 9, k, k 0 7,, 5, k, k 0 4, 6 RUGBY For Eercises 6 and 6, use the following information. The length of a Rugby League field is 5 meters longer than its width w. 6. Write an epression for the area of the rectangular field. [w(w 5)] m 6. The area of a Rugby League field is 860 square meters. Find the dimensions of the field. 0 m by 68 m Lesson 9-3 Factoring Trinomials: b c 493 NAME DATE PERIOD 9-3 Enrichment, p. 540 Puzzling Primes A prime number has only two factors, itself and. The number 6 is not prime because it has and 3 as factors; 5 and 7 are prime. The number is not considered to be prime.. Use a calculator to help you find the 5 prime numbers less than 00., 3, 5, 7,, 3, 7, 9, 3, 9, 3, 37 4, 43, 47, 53, 59, 6, 67, 7, 73, 79, 83, 89, 97 Prime numbers have interested mathematicians for centuries. They have tried to find epressions that will give all the prime numbers, or only prime numbers. In the 700s, Euler discovered that the trinomial 4 will yield prime numbers for values of from 0 through 39.. Find the prime numbers generated by Euler s formula for from 0 through 7. 4, 43, 47, 53, 6, 7, 83, 97 (h 6) cm WEB DESIGN For Eercises 34 and 35, use the following information. Janeel has a 0-inch by -inch photograph. She wants to scan the photograph, then reduce the result by the same amount in each dimension to post on her Web site. Janeel wants the area of the image to be one eighth that of the original photograph. 34. Write an equation to represent the area of the reduced image. (0 )( ) 5, or Find the dimensions of the reduced image. 3 in. by 5 in. Reading to Learn Mathematics, p NAME DATE PERIOD Reading to Learn Mathematics Factoring Trinomials: b c Pre-Activity How can factoring be used to find the dimensions of a garden? Read the introduction to Lesson 9-3 at the top of page 489 in your tetbook. Why do you need to find two numbers whose product is 54? The problem asks you to find the length and width of the garden. You know the area is 54 ft.since you multiply length times width to find area, you need to find two numbers whose product is 54. Why is the sum of these two numbers half the perimeter or 5? You add to find the perimeter of the garden. Since you use the length twice and the width twice to find the perimeter, the sum of the length and the width is half the perimeter or 5. Reading the Lesson Tell what sum and product you want m and n to have to use the pattern ( m)( n) to factor the given trinomial sum: 0 product: 4. 0 sum: product: 3. 4 sum: 4 product: sum: 6 product: 5. To factor 8 3, you can look for numbers with a product of 3 and a sum of 8. Eplain why the numbers in the pair you are looking for must both be negative. To have a product of positive 3, the numbers must both be positive or both be negative. If both were positive, their sum would be positive instead of negative. Helping You Remember 6 6. If you are using the pattern ( m)( n) to factor a trinomial of the form b c, how can you use your knowledge of multiplying integers to help you remember whether m and n are positive or negative factors? Sample answer: Like signs multiplied together result in positive numbers and unlike signs result in negative numbers. So, if c is positive, the factors m and n both have the same sign as b. If c is negative, one of the factors is positive and the other is negative. 0 ELL Lesson 9-3 Lesson 9-3 Factoring Trinomials: b c 493

29 4 Assess Open-Ended Assessment Speaking Ask volunteers to brainstorm a mnemonic device that will help them remember how to factor trinomials with different positive and negative values of b and c. Then write eample trinomials on the chalkboard and have students factor them using the mnemonic devices as a guide. Getting Ready for Lesson 9-4 PREREQUISITE SKILL Students will learn to factor additional types of trinomials in Lesson 9-4 using factoring by grouping. Use Eercises to determine your students familiarity with factoring by grouping. Assessment Options Quiz (Lesson 9-3) is available on p. 573 of the Chapter 9 Resource Masters. Mid-Chapter Test (Lessons 9- through 9-3) is available on p. 575 of the Chapter 9 Resource Masters. Answer 63. Answers should include the following. You would use a guess-andcheck process, listing the factors of 54, checking to see which pairs added to 5. To factor a trinomial of the form a c, you also use a guess-and-check process, list the factors of c, and check to see which ones add to a. Graphing Calculator Maintain Your Skills 70. 3, 5 SOL/EOC Practice Standardized Test Practice Mied Review 78. (y 3)(3y ) 79. (a 4)(3a ) 80. ( )(4 3) Getting Ready for the Net Lesson 494 Chapter 9 Factoring 63. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can factoring be used to find the dimensions of a garden? Include the following in your answer: a description of how you would find the dimensions of the garden, and an eplanation of how the process you used is related to the process used to factor trinomials of the form b c. 64. Which is the factored form of 7 4? C A ( )(y 4) B ( )( ) C ( 3)( 4) D ( 6)( 7) 65. GRID IN What is the positive solution of p 3p 30 0? 5 Use a graphing calculator to determine whether each factorization is correct. Write yes or no. If no, state the correct factorization ( 6)( 8) ( 5)( ) yes ( 33)( ) ( 0)( ) 66. no; ( 6)( 8) 68. no; ( )( 3) 69. no; ( 0)( ) Solve each equation. Check your solutions. (Lesson 9-) 70. ( 3)( 5) 0 7. b(7b 4) 0 0, y 9y 9 5, 0 Find the GCF of each set of monomials. (Lesson 9-) 73. 4, 36, p q 5, p 3 q 3 3p q y 5, 0 y 7, 75 3 y 4 5 y 4 INTERNET For Eercises 76 and 77, use the graph at the right. USA TODAY Snapshots (Lessons 3-7 and 8-3) 76. Find the percent increase in the number of domain registrations from 997 to % 77. Use your answer from Eercise 76 to verify the claim that registrations grew more than 8-fold from 997 to 000 is correct. (.54) 7.3(.54) ( 7.3)(.54) or 8.3(.54) Number of domain registrations climbs During the past four years,.com,.net and.org domain registrations have grown more than 8-fold: Source: Network Solutions (VeriSign) By Cindy Hall and Bob Laird, USA TODAY PREREQUISITE SKILL Factor each polynomial. (To review factoring by grouping, see Lesson 9-.) 78. 3y y 9y a a a p 6p 7p 8. 3b 7b b g g 6g 3 (p 7)(p 3) (b 4)(3b 7) (g 3)(g ) 8. million 9 million million million Online Lesson Plans USA TODAY Education s Online site offers resources and interactive features connected to each day s newspaper. Eperience TODAY, USA TODAY s daily lesson plan, is available on the site and delivered daily to subscribers. This plan provides instruction for integrating USA TODAY graphics and key editorial features into your mathematics classroom. Log on to Chapter 9 Factoring

30 Factoring Trinomials: a a Virginia SOL b c STANDARD A. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. Lesson Notes Vocabulary prime polynomial Study Tip Look Back To review factoring by grouping, see Lesson 9-. Factor trinomials of the form a b c. Solve equations of the form a b c 0. can algebra tiles be used to factor 7 6? The factors of 7 6 are the dimensions of the rectangle formed by the algebra tiles shown below. The process you use to form the rectangle is the same mental process you can use to factor this trinomial algebraically. FACTOR a b c For trinomials of the form b c, the coefficient of is. To factor trinomials of this form, you find the factors of c whose sum is b. We can modify this approach to factor trinomials whose leading coefficient is not. F O I L ( 5)(3 ) Use the FOIL method Observe the following pattern in this product a m n c a b c 5 7 and m n b and mn ac You can use this pattern and the method of factoring by grouping to factor Find two numbers, m and n, whose product is 6 5 or 30 and whose sum is 7. Factors of 30 Sum of Factors, 30 3, 5 7 The correct factors are and m n 5 Write the pattern m and n 5 (6 ) (5 5) Group terms with common factors. (3 ) 5(3 ) Factor the GCF from each grouping. (3 )( 5) 3 is the common factor. Therefore, (3 )( 5). Lesson 9-4 Factoring Trinomials: a b c 495 Focus 5-Minute Check Transparency 9-4 Use as a quiz or review of Lesson 9-3. Mathematical Background notes are available for this lesson on p. 47D. can algebra tiles be used to factor 7 6? Ask students: When you form a rectangle with algebra tiles to model a trinomial such as the one given, how do you know the factors? The length of the rectangle is one factor, and the height is the other factor. How is the trinomial 7 6 different from those that you learned how to factor in Lesson 9-3? The term is multiplied by a constant (). What would the rectangle formed by the given algebra tiles look like? What are the factors of the trinomial? ( 3)( ) Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 543 Practice, p. 544 Reading to Learn Mathematics, p. 545 Enrichment, p. 546 Graphing Calculator and Spreadsheet Masters, p. 39 Parent and Student Study Guide Workbook, p. 7 Prerequisite Skills Workbook, pp. 3 4 Transparencies 5-Minute Check Transparency 9-4 Answer Key Transparencies Technology AlgePASS: Tutorial Plus, Lesson 6 Interactive Chalkboard Lesson - Lesson Title 495

31 Teach FACTOR a b c In-Class Eamples a. Factor (5 )( 5) Teaching Tip In Eample b of the Student Edition, mn is a large number, which has quite a few factors. Have students start listing the factors that can easily be determined by mental math. More than likely, the factors that equal m n can be found this way, without too much calculation. b. Factor 4 3. (4 3)(6 ) Teaching Tip Remind students not to forget to record the common factor that they factored out of the trinomial when they record the other two factors. Factor ( )( 4) Power Point Study Tip Finding Factors Factor pairs in an organized list so you do not miss any possible pairs of factors. Eample Factor a b c a. Factor 7 3. In this trinomial, a 7, b and c 3. You need to find two numbers whose sum is and whose product is 7 3 or. Make an organized list of the factors of and look for the pair of factors whose sum is. Factors of Sum of Factors, The correct factors are and m n 3 Write the pattern. 7 3 m and n (7 ) ( 3) Group terms with common factors. (7 ) 3(7 ) Factor the GCF from each grouping. (7 )( 3) Distributive Property CHECK You can check this result by multiplying the two factors. F O I L (7 )( 3) 7 3 FOIL method 7 3 Simplify. b. Factor In this trinomial, a 0, b 43 and c 8. Since b is negative, m n is negative. Since c is positive, mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 0 8 or 80, and look for the pair of factors whose sum is 43. Factors of 80 Sum of Factors, 80 8, , , , , The correct factors are 8 and m n 8 Write the pattern. 0 ( 8) ( 35) 8 m 8 and n 35 (0 8) ( 35 8) Group terms with common factors. (5 4) 7( 5 4) Factor the GCF from each grouping. (5 4) 7( )(5 4) 5 4 ( )(5 4) (5 4) ( 7)(5 4) 7( ) 7 (5 4)( 7) Distributive Property 496 Chapter 9 Factoring Sometimes the terms of a trinomial will contain a common factor. In these cases, first use the Distributive Property to factor out the common factor. Then factor the trinomial. Eample Factor When a, b, and c Have a Common Factor Factor Notice that the GCF of the terms 3, 4, and 45 is 3. When the GCF of the terms of a trinomial is an integer other than, you should first factor out this GCF ( 8 5) Distributive Property Differentiated Instruction Interpersonal Place students in groups to factor polynomials such as those in Eamples and. Have each group member find one or two factors for mn, depending on the number of factors and number of students in the group. By dividing the labor, students should be able to quickly find the factors for mn that sum to m n. Once they find the factors, have students complete the factoring as a group. 496 Chapter 9 Factoring

32 Study Tip Factoring Completely Always check for a GCF first before trying to factor a trinomial. Now factor 8 5. Since the lead coefficient is, find two factors of 5 whose sum is 8. Factors of 5 Sum of Factors, 5 6 3, 5 8 The correct factors are 3 and 5. So, 8 5 ( 3)( 5). Thus, the complete factorization of is 3( 3)( 5). A polynomial that cannot be written as a product of two polynomials with integral coefficients is called a prime polynomial. Eample 3 Determine Whether a Polynomial Is Prime Factor 5. In this trinomial, a, b 5 and c. Since b is positive, m n is positive. Since c is negative, mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of or 4, where one factor in each pair is negative. Look for a pair of factors whose sum is 5. Factors of 4 Sum of Factors, 4 3, 4 3, 0 There are no factors whose sum is 5. Therefore, 5 cannot be factored using integers. Thus, 5 is a prime polynomial. In-Class Eample 3 4 Teaching Tip Make sure students list all possible factors of mn, including both positive and negative factors, before they decide the polynomial is prime. Factor prime SOLVE EQUATIONS BY FACTORING In-Class Eamples Power Point Power Point Teaching Tip Remind students that when a polynomial has two factors, there are two solutions. Check each solution by substituting it into the original equation. Solve 8b 9b 8 3b 5b. 4, 5 3 SOLVE EQUATIONS BY FACTORING Some equations of the form a b c 0 can be solved by factoring and then using the Zero Product Property. Eample 4 Solve Equations by Factoring Solve 8a 9a 5 4 3a. Check your solutions. 8a 9a 5 4 3a Original equation 8a 6a 9 0 Rewrite so that one side equals 0. (4a 3)(a 3) 0 Factor the left side. 4a 3 0 or a 3 0 Zero Product Property 4a 3 a 3 Solve each equation. a 3 4 a 3 The solution set is 3 4, 3. CHECK Check each solution in the original equation. 8a 9a 5 4 3a a 9a 5 4 3a MODEL ROCKETS Ms. Nguyen s science class built an air-launched model rocket for a competition. When they test-launched their rocket outside the classroom, the rocket landed in a nearby tree. If the launch pad was feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation h 6t vt s. 3.5 seconds Lesson 9-4 Factoring Trinomials: a b c 497 Unlocking Misconceptions Dividing Students may assume that because they cannot divide each side of ( ) 0 by, they may never divide both sides of an equation set equal to zero by anything. Make sure they realize that they can divide each side by a known number, as in Eample 5, but they cannot divide each side by a variable that may equal zero. Lesson 9-4 Factoring Trinomials: a b c 497

33 3 Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 9. include eplanations on how to factor trinomials of the form a b c. include any other item(s) that they find helpful in mastering the skills in this lesson. Study Tip Factoring When a Is Negative When factoring a trinomial of the form a b c, where a is negative, it is helpful to factor out a negative monomial. A model for the vertical motion of a projected object is given by the equation h 6t vt s, where h is the height in feet, t is the time in seconds, v is the initial upward velocity in feet per second, and s is the starting height of the object in feet. Eample 5 Solve Real-World Problems by Factoring PEP RALLY At a pep rally, small foam footballs are launched by cheerleaders using a sling-shot. How long is a football in the air if a student in the stands catches it on its way down 6 feet above the gym floor? Use the model for vertical motion. h 6t vt s Vertical motion model 6 6t 4t 6 h 6, v 4, s 6 0 6t 4t 0 Subtract 6 from each side. 0 (8t t 0) Factor out. 0 8t t 0 Divide each side by. 0 (8t 5)(t ) Factor 8t t 0. 8t 5 0 or t 0 Zero Product Property 8t 5 t Solve each equation. t 5 8 Height of release 6 ft t 0 v 4 ft/s Height of reception 6 ft FIND THE ERROR Students should first look at the numbers for which Dasan and Craig are finding factors. This clue should immediately tell them which student is correct. Then, challenge students to find another error in Dasan s work. Students should notice that he factored a out of 8, which is not possible. About the Eercises Organization by Objective Factor a b c: 4 34 Solve Equations by Factoring: Odd/Even Assignments Eercises 4 48 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 5 9 odd, odd, 49, 50, Average: 5 47 odd, 49 5, Advanced: 4 48 even, 5 6 (optional: 63 70) All: Practice Quiz ( 0) Concept Check. m and n are the factors of ac that add to b. GUIDED PRACTICE KEY Eercises Eamples Chapter 9 Factoring Answers Guided Practice 3. When factoring a trinomial of the form a b c, where a, you must find the factors of ac, not of c. 4. ( 5)( ) 5. (3 )( ) The solutions are 5 8 second and seconds. The first time represents how long it takes the football to reach a height of 6 feet on its way up. The later time represents how long it takes the ball to reach a height of 6 feet again on its way down. Thus, the football will be in the air for seconds before the student catches it.. Eplain how to determine which values should be chosen for m and n when factoring a polynomial of the form a b c.. OPEN ENDED Write a trinomial that can be factored using a pair of numbers whose sum is 9 and whose product is 4. Sample answer: FIND THE ERROR Dasan and Craig are factoring 8. Dasan Factors of 8 Sum, 8 9 3, 6 9 9, = ( + + 8) = ( + 9)( + ) Craig Factors of 36 Sum, 36 37, 8 0 3, 5 4, 9 3 6, is prime. Who is correct? Eplain your reasoning. Craig; see margin for eplanation. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. 4. (3a )(a ) 6. (p 3)(p 4) 4. 3a 8a 4 5. a a 7 prime 6. p 4p n 4n 35 ( 4)( 5) 3( )( 3) (n 5)(n 7) 6. (p 3)(3p ) 7. (5d 4)(d ) 8. prime 9. (3g )(3g ) 0. (a 3)(a 6). ( 4)( 5). (5c 7)(c ) 3. prime 4. (y 3)(4y 3) 5. (5n )(n 3) 6. (5z 9)(3z ) 7. ( 3)(7 4) 8. (3r )(r 3) 9. 5(3 )( 3) 30. (3 5y)(3 5y) 3. (a 5b)(3a b) 498 Chapter 9 Factoring

34 0. 3, 3 Application indicates increased difficulty Practice and Apply For See Eercises Eamples Etra Practice See page 840. Solve each equation. Check your solutions p 9p n 7n 0 5, 4 3, GYMNASTICS When a gymnast making a vault leaves the horse, her feet are 8feet above the ground traveling with an initial upward velocity of 8 feet per second. Use the model for vertical motion to find the time t in seconds it takes for the gymnast s feet to reach the mat. (Hint: Let h 0, the height of the mat.) s Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime See margin p 5p d 6d k 9k g g 4 0. a 9a c 7c p 5p y 6y n n z 7z r 4r y 5y 3. 36a 9ab 0b CRITICAL THINKING Find all values of k so that each trinomial can be factored as two binomials using integers. 3. k 33. k k, k 0 5, 4,, 0 3, 7, 3, 0, 6, 8 Solve each equation. Check your solutions See p. 5A n 5n a 3a t 3 t (3y )(y 3) y (4a )(a ) 7a 5 GEOMETRY For Eercises 49 and 50, use the following information. Arectangle with an area of 35 square inches is formed by cutting off strips of equal width from a rectangular piece of paper. 49. Find the width of each strip. in. 50. Find the dimensions of the new rectangle. 5 in. by 7 in. 5. CLIFF DIVING Suppose a diver leaps from the edge of a cliff 80 feet above the ocean with an initial upward velocity of 8 feet per second. How long will it take the diver to enter the water below?.5 s Lesson 9-4 Factoring Trinomials: a b c ft NAME DATE PERIOD 9-4 Enrichment, p. 546 Area Models for Quadratic Trinomials After you have factored a quadratic trinomial, you can use the factors to draw geometric models of the trinomial. 5 6 ( )( 6) To draw a rectangular model, the value was used for so that the shorter side would have a length of. Then the drawing was done in centimeters. So, the area of the rectangle is 5 6. To draw a right triangle model, recall that the area of a triangle is one-half the base times the height. So, one of the sides must be twice as long as the shorter side of the rectangular model. 5 6 ( )( 6) ( )( 6) The area of t right triangle is also ft/s 7 in in. Study Study Guide and and Intervention Intervention, p. 54 (shown) and p NAME DATE PERIOD Factoring Trinomials: a b c Factor a b c To factor a trinomial of the form a b c, find two integers, m and n whose product is equal to ac and whose sum is equal to b. If there are no integers that satisfy these requirements, the polynomial is called a prime polynomial. Eample Factor 5 8. Eample Factor In this eample, a, b 5, and c 8. Note that the GCF of the terms 3,3, You need to find two numbers whose sum is and 8 is 3. First factor out this GCF. 5 and whose product is 8 or 36. Make 3 a list of the factors of 36 and look for the 3 8 3( 6). pair of factors whose sum is 5. Now factor 6. Since a, find the two factors of 6 whose sum is. Factors of 36 Sum of Factors Factors of 6 Sum of Factors, 36 37, 6 5, 8 0, 6 5 3, 5, 3 Use the pattern a m n c, with, 3 a, m 3, n, and c Now use the pattern ( m)( n) with ( 3) ( 8) m and n 3. ( 3) 6( 3) 6 ( )( 3) ( 6)( 3) The complete factorization is Therefore, ( 6)( 3) ( )( 3). Eercises Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime m 8m r 8r ( )( ) (3m )(m 3) (4r )(4r ) ( 3)(3 ) (3 4)( ) (3 5)(6 ) 7. a 5a y 9y c 9c (a 3)(a ) (6y 5)(3y ) (4c 7)(3 c ) p 60p (4 8)( 3) (p 5)(4p 5) (6 5)(8 3) 3. 3y 6y m 44m 48 3(y )(y 4) ( 8)( 3) 4(m 3)(m 4) a 4a y y prime (a 7a 9) prime Skills Practice, p. 543 and NAME DATE PERIOD 9-4 Practice (Average) Practice, Factoring Trinomials: p. 544 a (shown) b c Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime.. b 0b. 3g 8g (b )(b 3) (3g )(g ) ( 3)( ) 4. 8b 5b m 7m d 7d 0 prime (3m )(m 3) (5d 4)(d 5) 7. 6a 7a 8. 8w 8w (3a 4)(a 3) (4w 3)(w 3) prime 0. 5n n r 5r 6 (5n 7)(3n 4) ( 5)(5 ) 3(3r )(r ) 3. y 4y k 9k z 0z 48 (y )(6y 5) (k 3)(7k 6) 4(z 4)(z 3) 6. q 34q h 5h 8 8. p p 0 (3q )(q 7) 3(h 3)(3h ) (3p )(p 5) Solve each equation. Check your solutions. 9. 3h h n n. 8q 0q 3 0 8,,, 3. 6b 5b c c 4c g 0 9g, 4 3,, y 7y 6. 9z 6z 5 7. k 5k 6k 0, 5, 5, a 6a 6a 30. 8a 0a a 4, 3, 4, 3. DIVING Lauren dove into a swimming pool from a 5-foot-high diving board with an initial upward velocity of 8 feet per second. Find the time t in seconds it took Lauren to enter the water. Use the model for vertical motion given by the equation h 6t vt s, where h is height in feet, t is time in seconds, v is the initial upward velocity in feet per second, and s is the initial height in feet. (Hint: Let h 0 represent the surface of the pool.).5 s 3. BASEBALL Brad tossed a baseball in the air from a height of 6 feet with an initial upward velocity of 4 feet per second. Enrique caught the ball on its way down at a point 4 feet above the ground. How long was the ball in the air before Enrique caught it? Use the model of vertical motion from Eercise 3. s Reading to Learn Mathematics, p NAME DATE PERIOD Reading to Learn Mathematics Factoring Trinomials: a b c ELL Pre-Activity How can algebra tiles be used to factor 7 6? Read the introduction to Lesson 9-4 at the top of page 495 in your tetbook. When you form the algebra tiles into a rectangle, what is the first step? Place the two tiles on the product mat and arrange the si tiles into a rectangular array. What is the second step? Arrange the seven tiles to complete the rectangle. Reading the Lesson. Suppose you want to factor the trinomial a. What is the first step? Find integers with a product of 4 and a sum of 4. The integers are and. b. What is the second step? Rewrite the polynomial by breaking the middle term into two addends that use and as coefficients. You can use c. Provide an eplanation for the net two steps. (3 ) ( 8) Group terms with common factors. (3 ) 4(3 ) Factor the GCF from each grouping. d. Use the Distributive Property to rewrite the last epression in part c. You get ( 4 )(3 ).. Eplain how you know that the trinomial 7 4 is a prime polynomial. To factor 7 4, you would need two negative integers whose product is 8 and whose sum is 7. There are no such negative integers. Therefore, 7 4 is prime. Helping You Remember 3. What are steps you could use to remember how to find the factors of a trinomial written in the form of a b c? Sample answer: Look for two integers with a product equal to ac and a sum of b. Replace the middle term with a sum of terms that have those two integers as coefficients. Then factor by grouping. Lesson 9-4 Lesson 9-4 Lesson 9-4 Factoring Trinomials: a b c 499

35 4 Assess Open-Ended Assessment Writing Place students in pairs. Have each student pick a problem that involves solving an equation by factoring and intentionally make a mistake while solving the problem. Then have students echange problems. Students should write a description of what is incorrect, and how to solve the problem correctly. Have students share their eplanations and correct solutions with the class. Getting Ready for Lesson 9-5 PREREQUISITE SKILL Students will learn to factor binomials that are differences of squares in Lesson 9-5. Factoring differences of squares requires students to be able to quickly find the principal square roots of perfect squares. Use Eercises to determine your students familiarity with finding square roots. Assessment Options Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 9-3 and 9-4. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not yet mastered. Answer 53. You can use algebra tiles to factor 7 6 by finding the dimensions of the rectangle that is formed by the tiles for 7 6. Answers should include the following. 3 by With algebra tiles, you can try various ways to make a rectangle with the necessary tiles. Once you make the rectangle, however, the dimensions of the rectangle are the factors of the polynomial. In a way, you have to go through the guess-and-check process whether you are factoring algebraically or geometrically (using algebra tiles.) Maintain Your Skills , 4 P SOL/EOC Practice Standardized Test Practice Mied Review 60. 9, 3 Getting Ready for the Net Lesson ractice Quiz Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lessons 9-3 and 9-4). 4 7 ( 4)( 8). 8p 6p 35 (p 5)(4p 7) 3. 6a 4a 5 (4a )(4a 5) 4. n 7n 5 (n 3)(n 4) 5. 4c 6c y 33y 54 3(y )(y 9) (3c )(4c 9) Solve each equation. Check your solutions. (Lessons 9-3 and 9-4) 7. b 4b 3 0 { 6, } {3, 5} 9. y 7y 0 3 4, a 5a 4 3, Chapter 9 Factoring Guess ( )( 3) incorrect because 8 tiles are needed to complete the rectangle 5. CLIMBING Damaris launches a grappling hook from a height of 6 feet with an initial upward velocity of 56 feet per second. The hook just misses the stone ledge of a building she wants to scale. As it falls, the hook anchors on the ledge, which is 30 feet above the ground. How long was the hook in the air? 3 s 53. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can algebra tiles be used to factor 7 6? Include the following in your answer: the dimensions of the rectangle formed, and an eplanation, using words and drawings, of how this geometric guess-and-check process of factoring is similar to the algebraic process described on page What are the solutions of p p 3 0? D A 3 and B 3 and C 3 and D 3 and 55. Suppose a person standing atop a building 398 feet tall throws a ball upward. If the person releases the ball 4 feet above the top of the building, the ball s height h, in feet, after t seconds is given by the equation h 6t 48t 40. After how many seconds will the ball be 338 feet from the ground? B A 3.5 B 4 C 4.5 D 5 Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-3) 56. a 4a 57. t t 58. d 5d 44 (a 3)(a 7) prime (d 4)(d ) Solve each equation. Check your solutions. (Lesson 9-) 59. (y 4)(5y 7) (k 9)(3k ) 0 6. u u {0, } 6. BUSINESS Jake s Garage charges $83 for a two-hour repair job and $85 for a five-hour repair job. Write a linear equation that Jake can use to bill customers for repair jobs of any length of time. (Lesson 5-3) y 34 5 PREREQUISITE SKILL Find the principal square root of each number. (To review square roots, see Lesson -7.) Lessons 9-3 and Chapter 9 Factoring

36 Study Tip Look Back To review the product of a sum and a difference, see Lesson 8-8. Factoring Differences a Virginia SOL of Squares Factor binomials that are the differences of squares. Solve equations involving the differences of squares. A basketball player s hang time is the length of time he is in the air after jumping. Given the maimum height h a player can jump, you can determine his hang time t in seconds by solving 4t h 0. If h is a perfect square, this equation can be solved by factoring using the pattern for the difference of squares. FACTOR a b of squares. Difference of Squares Use a straightedge to draw two squares similar to those shown below. Choose any measures for a and b. Notice that the area of the large square is a, and the area of the small square is b. Cut the irregular region into two congruent pieces as shown below. a a a b Region b b a STANDARD A. The student will factor completely firstand second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. can you determine a basketball player s hang time? Ageometric model can be used to factor the difference b b a b Region a Cut the small square from the large square. a b a b a The area of the remaining irregular region is a b. Rearrange the two congruent regions to form a rectangle with length a b and width a b. b b a a b a b Region a b a b Region Make a Conjecture. Write an epression representing the area of the rectangle. (a b)(a b). Eplain why a b = (a b)(a b). Since a b and (a b)(a b) describe the same area, a b (a b)(a b). Lesson 9-5 Factoring Differences of Squares 50 a b b b Lesson Notes Focus 5-Minute Check Transparency 9-5 Use as a quiz or review of Lesson 9-4. Mathematical Background notes are available for this lesson on p. 47D. Building on Prior Knowledge In Chapter 8, students learned about a special binomial product called a difference of squares. Specifically, a difference of squares is the product of two binomials of the form (a b)(a b), and the product is of the form a b. In this lesson, students will learn to factor differences of squares, and use the factoring of differences of squares to solve equations. can you determine a basketball player s hang time? Ask students: What is a perfect square? A perfect square is a rational number whose square root is a rational number. Is 4t a perfect square? If so, what is its principal square root? Yes, the principal square root is t. If a basketball player can jump 4 feet, what would be her hang time? second Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 549 Practice, p. 550 Reading to Learn Mathematics, p. 55 Enrichment, p. 55 Assessment, p. 574 Prerequisite Skills Workbook, pp. 3 4 Graphing Calculator and Spreadsheet Masters, p. 40 Parent and Student Study Guide Workbook, p. 7 School-to-Career Masters, p. 8 Science and Mathematics Lab Manual, pp Teaching Algebra With Manipulatives Masters, pp. 4, 67 Transparencies 5-Minute Check Transparency 9-5 Answer Key Transparencies Technology Interactive Chalkboard Lesson - Lesson Title 50

37 Teach FACTOR a b In-Class Eamples Factor each binomial. a. m 64 (m 8)(m 8) 3 4 Teaching Tip Students may check their factoring by multiplying the factors using the FOIL method. The first-degree term will always drop out when the product is a difference of squares. b. 6y 8z (4y 9z)(4y 9z) Factor 3b 3 7b. 3b(b 3)(b 3) Teaching Tip Students should notice that when the difference of squares factoring technique has been applied once, one of the factors should be prime. Factor 4y (y 5)(y 5)(y 5) Power Point Factor ( )( )( 5) Study Tip Common Misconception Remember that the sum of two squares, like 9, is not factorable using the difference of squares pattern. 9 is a prime polynomial. Symbols a b (a b)(a b) or (a b)(a b) Eample 9 ( 3)( 3) or ( 3)( 3) We can use this pattern to factor binomials that can be written in the form a b. Eample Factor the Difference of Squares Factor each binomial. a. n 5 n 5 n 5 Write in the form a b. (n 5)(n 5) Factor the difference of squares. b y 36 49y (6) (7y) Eample (6 7y) (6 7y) and 49y 7y 7y Factor the difference of squares. Factor Out a Common Factor Factor 48a 3 a. 48a 3 a a(4a ) The GCF of 48a 3 and a is a. a[(a) ] 4a a a and Eample 3 a(a )(a ) Factor the difference of squares. Difference of Squares If the terms of a binomial have a common factor, the GCF should be factored out first before trying to apply any other factoring technique. Occasionally, the difference of squares pattern needs to be applied more than once to factor a polynomial completely. Apply a Factoring Technique More Than Once Factor ( 4 8) The GCF of 4 and 6 is. [( ) 9 ] 4 and ( 9)( 9) Factor the difference of squares. ( 9)( 3 ) and ( 9)( 3)( 3) Factor the difference of squares. Eample 4 Apply Several Different Factoring Techniques Factor Original polynomial 5( 3 3 3) Factor out the GCF. 5[( 3 ) (3 3)] Group terms with common factors. 5[( ) 3( )] Factor each grouping. 5( )( 3) is the common factor. 5( )( )( 3) Factor the difference of squares,, into ( )( ). 50 Chapter 9 Factoring Algebra Activity Materials: straightedge, scissors Using graph paper, students are more likely to draw straight squares, which will make the final product appear more like a rectangle. Make sure students label their figures as shown. Eplain that the sides of the original square have length of a, and when the b square is cut out, the remaining sides have lengths of a b. 50 Chapter 9 Factoring

38 Study Tip Alternative Method The fraction could also be cleared from the equation in Eample 5a by multiplying each side of the equation by 6. p p 9 0 (4p 3)(4p 3) 0 4p 3 0 or 4p 3 0 p 3 4 p 3 4 SOLVE EQUATIONS BY FACTORING You can apply the Zero Product Property to an equation that is written as the product of any number of factors set equal to 0. Eample 5 Solve Equations by Factoring Solve each equation by factoring. Check your solutions. a. p p 9 0 Original equation 6 p p p p and p 3 4 p Factor the difference of squares. p or p Zero Product Property p 3 4 p 3 4 Solve each equation. The solution set is 3 4, 3 4. Check each solution in the original equation. b Original equation Subtract 50 from each side. (9 5) 0 The GCF of 8 3 and 50 is. (3 5)(3 5) and Applying the Zero Product Property, set each factor equal to 0 and solve the resulting three equations. 0 or or SOLVE EQUATIONS BY FACTORING In-Class Eamples 6 Teaching Tip Students may not be used to thinking of fractions as perfect squares. Remind them that if both the numerator and denominator are perfect squares, then the fraction itself is a perfect square. Also, if students are uncomfortable finding square roots of fractions, point out the Study Tip in the tet. 5 Solve each equation by factoring. Check your solutions. a. q 4 0, b. 48y 3 3y, 0, 4 4 Power Point EXTENDED RESPONSE A square with side length is cut from the right triangle shown below. SOL/EOC Practice Standardized Test Practice Test-Taking Tip Look to see if the area of an oddly-shaped figure can be found by subtracting the areas of more familiar shapes, such as triangles, rectangles, or circles. The solution set is 5 3, 0, 5 3. Check each solution in the original equation. Eample 6 Use Differences of Two Squares Etended-Response Test Item Acorner is cut off a -inch by -inch square piece of paper. The cut is inches from a corner as shown. a. Write an equation in terms of that represents the area A of the paper after the corner is removed. b. What value of will result in an area that is 7 the 9 area of the original square piece of paper? Show how you arrived at your answer. Read the Test Item A is the area of the square minus the area of the triangular corner to be removed. (continued on the net page) Lesson 9-5 Factoring Differences of Squares 503 in. in. 6 8 a. Write an equation in terms of that represents the area A of the figure after the corner is removed. A 64 b. What value of will result in 3 a figure that is of the area 4 of the original triangle? Show how you arrived at your answer. 4; solve Standardized Test Practice Eample 6 Eplain to students that when they complete etended response test items, it is very important to show all the steps in working out the problem, rather than just writing the answer. Often, full credit is not given for an etended response item unless all of the steps are shown. Conversely, students may receive partial credit for an incorrect answer, if their work shows only an arithmetic error, but the problem was solved using the correct method. Lesson 9-5 Factoring Differences of Squares 503

39 3 Practice/Apply Study Notebook Have students add the definitions/eamples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 9. include eplanations on how to factor differences of squares. include any other item(s) that they find helpful in mastering the skills in this lesson. Solve the Test Item a. The area of the square is or 4 square inches, and the area of the triangle is or square inches. Thus, A 4. b. Find so that A is 7 9 the area of the original square piece of paper, A o. A 7 9 A o Translate the verbal statement (4) A 4 and A o is Simplify Subtract from each side Simplify Multiply each side by 8 to remove fractions. (4 3)(4 3) 0 Factor the difference of squares or Zero Product Property Solve each equation. 3 Since length cannot be negative, the only reasonable solution is 4 3. FIND THE ERROR Make sure students can eplain what Jessica did wrong. Stress that Jessica s error is a common one. Ask students what they can do to avoid making the same mistake themselves. About the Eercises Organization by Objective Factor a b : 6 33 Solve Equations by Factoring: Odd/Even Assignments Eercises 6 45 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 7 3 odd, odd, 46, 47, 49, 5 70 Average: 7 43 odd, 46, 47, 49, 5 70 Advanced: 6 50 even, 5 64 (optional: 65 70) Concept Check. The binomial is the difference of two terms, each of which is a perfect square. 3. Yes; 3n 48 3(n 6) 3(n 4)(n 4). Guided Practice GUIDED PRACTICE KEY Eercises Eamples Chapter 9 Factoring. Describe a binomial that is the difference of two squares.. OPEN ENDED Write a binomial that is the difference of two squares. Then factor your binomial. Sample answer: 5 ( 5)( 5) 3. Determine whether the difference of squares pattern can be used to factor 3n 48. Eplain your reasoning. 4. FIND THE ERROR Manuel and Jessica are factoring 64 6y. Manuel y = 6(4 + y ) Jessica y = 6(4 + y ) = 6( + y)( y) Who is correct? Eplain your reasoning. Manuel; 4 y is not the difference of squares. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 8. (4 y )( y)( y) 5. n 8 (n 9)(n 9) a ( 3a)( 3a) ( 7)( 7) y t 7 prime ( 3)( 3)( 3) Solve each equation by factoring. Check your solutions.. 4y 5 5, k 0, , 6 4. a 49a 3 7 Differentiated Instruction, 0, 7 Intrapersonal Consider having students complete the Check for Understanding problems on one day, and then check their own work on the net day, eamining their work and answers. Letting their own work sit for a day often allows students to see mistakes or problems in their work that they otherwise might not have noticed. It may also give students more of a chance to ask for help on difficult problems. 504 Chapter 9 Factoring

40 Standardized Test Practice SOL/EOC Practice 5. OPEN ENDED The area of the shaded part of the square at the right is 7 square inches. Find the dimensions of the square. in. by in. Study Study Guide and and Intervention Intervention, p. 547 (shown) and p NAME DATE PERIOD Factoring Differences of Squares Factor a b The binomial epression a b is called the difference of two squares.the following pattern shows how to factor the difference of squares. Difference of Squares a b (a b)(a b) (a b)(a b). indicates increased difficulty Practice and Apply For See Eercises Eamples Etra Practice See page 84.. ( 3r)( 3r). (0c d) (0c d) 4. (a 7b) (a 7b) 5. (3y 6z) (3y 6z) 6. (d 3)(d 3) 7. 3( 5)( 5) 9. (g 5) 30. 8a (a ) (a ) Aerodynamics Lift works on the principle that as the speed of a gas increases, the pressure decreases. As the velocity of the air passing over a curved wing increases, the pressure above the wing decreases, lift is created, and the wing rises. Source: Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 9. (5 p)(5 p) 0. (7h 4)(7h 4) ( 7)( 7) 7. n 36 (n 6)(n 6) k prime p h. 9r. 00c d y prime 4. 44a 49b 5. 69y 36z 6. 8d z 64 8(z 8) 9. 4g a 4 7a y 3. n 3 5n 4n (a b) c 5( 3y)( 3y) (n )(n )(n 5) (a b c)(a b c) Solve each equation by factoring. Check your solutions y n a w p r 0 { 8} { 0} 4. d 3 47d 0 7, 0, n 3 50n 0 5 3, 0, { 3,, } 3, 0, 3, CRITICAL THINKING Show that a b (a b)(a b) algebraically. (Hint: Rewrite a b as a ab ab b.) See margin. 47. BOATING The United States Coast Guard s License Eam includes questions dealing with the breaking strength of a line. The basic breaking strength b in pounds for a natural fiber line is determined by the formula 900c b, where c is the circumference of the line in inches. What circumference of natural line would have 3600 pounds of breaking strength? in. 48. AERODYNAMICS The formula for the pressure difference P above and below a wing is described by the formula P dv dv, where d is the density of the air, v is the velocity of the air passing above, and v is the velocity of the air passing below. Write this formula in factored form. P d(v v )(v v ) 49. LAW ENFORCEMENT If a car skids on dry concrete, police can use the formula s 4 d to approimate the speed s of a vehicle in miles per hour given the length d of the skid marks in feet. If the length of skid marks on dry concrete are 54 feet long, how fast was the car traveling when the brakes were applied? 36 mph 50. PACKAGING The width of a bo is 9 inches more than its length. The height of the bo is inch less than its length. If the bo has a volume of 7 cubic inches, what are the dimensions of the bo? 3 in. by in. by in. Lesson 9-5 Factoring Differences of Squares Eample Factor each binomial. Eample a. n 64 n 64 n 8 Write in the form a b. (n 8)(n 8) Factor. b. 4m 8n 4m 8n (m) (9n) Write in the form a b. (m 9n)(m 9n) Factor. a. 50a 7 Factor each polynomial. 50a 7 (5a 36) Find the GCF. [(5a) 6 )] 5a 5a 5a and (5a 6)(5a 6) Factor the difference of squares. b Original polynomial 4( 3 ) Find the GCF. 4[( 3 ) ( )] Group terms. 4[ ( ) ( )] Find the GCF. 4[( )( )] Factor by grouping. 4[( )( )( )] Factor the difference of squares. Eercises Factor each polynomial if possible. If the polynomial cannot be factored, write prime.. 8. m n 5 ( 9)( 9) (m 0)(m 0) (4n 5)(4n 5) y a 9b (6 0y)(6 0y) prime (4a 3b)(4a 3b) 7. 5c a 8. 7p (5c a)(5c a) (6p 5)(6p 5) ( )( ) 0. 8 a a. 8y 00 (a 3)(a 3)(a 9) 6( 3a)( 3a) 8(y 5)(y 5) y 4 3y 5. 8m 3 8m 4( 5)( 5) y (y 4)(y 4) 8m(m 4)(m 4) a 3 98ab 8. 8y 7y 4 prime a(a 7b)(a 7b) 8y ( y)( y) a 4 3a (3 )(3 ) 3a (a )(a ) 3( )( )( ) Skills Practice, p. 549 and NAME DATE PERIOD 9-5 Practice (Average) Practice, Factoring Differences p. 550 of (shown) Squares Factor each polynomial, if possible. If the polynomial cannot be factored, write prime.. k r 3. 6p 36 (k 0)(k 0) (9 r)(9 r) (4p 6)(4p 6) f 6. 36g 49h prime ( 3f)( 3f) (6g 7h)(6g 7h) 7. m 44n y 9. 4a 54b (m n)(m n) 8( y)( y) 6(a 3b)(a 3b) 0. 3s 8u. 9d 3. 36z 3 9z (4s 3u)(4s 3u) prime 9z(z )(z ) 3. 45q 3 0q 4. 00b 3 36b 5. 3t 4 48t 5q(3q )(3q ) 4b(5b 3)(5b 3) 3t (t 4)(t 4) Solve each equation by factoring. Check your solutions. 6. 4y p b k 0. s. v h 48h 4. 75g 47g 3 3 { 30} 4 3,0 7 5,0 5. EROSION A rock breaks loose from a cliff and plunges toward the ground 400 feet below. The distance d that the rock falls in t seconds is given by the equation d 6t. How long does it take the rock to hit the ground? 5 s 6. FORENSICS Mr. Cooper contested a speeding ticket given to him after he applied his brakes and skidded to a halt to avoid hitting another car. In traffic court, he argued that the length of the skid marks on the pavement, 50 feet, proved that he was driving under the posted speed limit of 65 miles per hour. The ticket cited his speed at 70 miles per hour. Use the formula s d, where s is the speed of the car and d is the length of the skid 4 marks, to determine Mr. Cooper s speed when he applied the brakes. Was Mr. Cooper correct in claiming that he was not speeding when he applied the brakes? 60 mi/h; yes Reading to Learn Mathematics, p NAME DATE PERIOD Reading to Learn Mathematics Factoring Differences of Squares ELL Pre-Activity How can you determine a basketball player s hang time? Read the introduction to Lesson 9-5 at the top of page 50 in your tetbook. Suppose a player can jump feet. Can you use the pattern for the difference of squares to solve the equation 4t 0? Eplain. No; is not a perfect square. Reading the Lesson. Eplain why each binomial is a difference of squares. a is the square of, 5 is the square of 5, and the operation sign is a minus sign. b. 49a 64b 49a is the square of 7a, 64b is the square of 8b, and the operation sign is a minus sign. Lesson 9-5. Sometimes it is necessary to apply more than one technique when factoring, or to apply the same technique more than once. a. What should you look for first when you are factoring a binomial? Look for a factor or factors common to the terms of the binomial. Answer 46. Use factoring by grouping. a b a ab ab b (a ab) (ab b ) a(a b) b(a b) (a b)(a b) NAME DATE PERIOD 9-5 Enrichment, p. 55 Factoring Trinomials of Fourth Degree Some trinomials of the form a 4 a b b 4 can be written as the difference of two squares and then factored. Eample Factor y 9y 4. Step Find the square roots of the first and last terms y 4 3y Step Find twice the product of the square roots. ( )(3y ) y Step 3 Separate the middle term into two parts. One part is either your answer to Step or its opposite. The other part should be the opposite of a perfect square. 37 y y 5 y Step 4 Rewrite the trinomial as the difference of two squares and then factor. 4 4 y 9y 4 (4 4 9y 4 ) b. Eplain what is done in each step to factor ( 4 6) Factor out the GCF. 4[( ) 4 ] Write 4 6 in difference of squares form. 4( 4)( 4) Factor the difference of squares. 4( 4)( ) Write 4 in difference of squares form. 4( 4)( )( ) Factor the difference of squares. 3. Suppose you are solving the equation and rewrite it as (4 3)(4 3) 0. What would be your net steps in solving the equation? Set each factor equal to zero, then solve the resulting equations. Helping You Remember 4. How can you remember whether a binomial can be factored as a difference of squares? The operation sign must be a minus sign, and the epressions before and after the minus sign must be perfect squares. Lesson 9-5 Lesson 9-5 Factoring Differences of Squares 505

41 4 Assess Open-Ended Assessment Modeling Have students complete an area problem similar to Eample 6 using grid paper. Have them demonstrate using the grid that the area and lengths they come up with using factoring a difference of squares is correct. Getting Ready for Lesson 9-6 PREREQUISITE SKILL Students will learn to factor perfect square trinomials in Lesson 9-6. When they learn how to factor perfect square trinomials, students will have to know how to square binomials. Use Eercises to determine your students familiarity with finding special products, such as the square of a binomial. 5. The flaw is in line 5. Since a b, a b 0. Therefore dividing by a b is dividing by zero, which is undefined. SOL/EOC Practice Standardized Test Practice 5. CRITICAL THINKING The following statements appear to prove that is equal to. Find the flaw in this proof. Suppose a and b are real numbers such that a b, a 0, b 0. () a b Given. () a ab Multiply each side by a. (3) a b ab b Subtract b from each side. (4) (a b)(a b) b(a b) Factor. (5) a b b Divide each side by a b. (6) a a a Substitution Property; a b (7) a a Combine like terms. (8) Divide each side by a. 5. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can you determine a basketball player s hang time? Include the following in your answer: a maimum height that is a perfect square and that would be considered a reasonable distance for a student athlete to jump, and a description of how to find the hang time for this maimum height. 53. What is the factored form of 5b? A A (5b )(5b ) B (5b )(5b ) C (5b )(5b ) D (5b )(b ) 54. GRID IN In the figure, the area between the two squares is 7 square inches. The sum of the perimeters of the two squares is 68 inches. How many inches long is a side of the larger square? 9 in. Assessment Options Quiz (Lessons 9-4 and 9-5) is available on p. 574 of the Chapter 9 Resource Masters. Answers 5. Answers should include the following. foot To find the hang time of a student athlete who attains a maimum height of foot, solve the equation 4t 0. You can factor the left side using the difference of squares pattern since 4t is the square of t and is the square of. Thus the equation becomes (t )(t ) 0. Using the Zero Product Property, each factor can be set equal to zero, resulting in two solutions, t and t. Since time cannot be negative, the hang time is second Maintain Your Skills Mied Review Getting Ready for the Net Lesson 506 Chapter 9 Factoring Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 9-4) 55. n 5n p 9p 0 prime ( )(3 4) (3p 5)(7p ) Solve each equation. Check your solutions. (Lesson 9-3) 58. y 8y k 8k 5 {3, 5} 60. b 8 b {, 4} { 6, } 6. STATISTICS Amy s scores on the first three of four 00-point biology tests were 88, 90, and 9. To get a B in the class, her average must be between 88 and 9, inclusive, on all tests. What score must she receive on the fourth test to get a B in biology? (Lesson 6-4) between 83 and 99, inclusive Solve each inequality, check your solution, and graph it on a number line. (Lesson 6-) See margin for graphs d d r r PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.) 65. ( )( ) 66. ( 6)( 6) 67. ( 8) 68. (3 4)(3 4) 69. (5 ) 70. (7 3) Chapter 9 Factoring

42 Reading Mathematics The Language of Mathematics Mathematics is a language all its own. As with any language you learn, you must read slowly and carefully, translating small portions of it at a time. Then you must reread the entire passage to make complete sense of what you read. In mathematics, concepts are often written in a compact form by using symbols. Break down the symbols and try to translate each piece before putting them back together. Read the following sentence. a ab b (a b) The trinomial a squared plus twice the product of a and b plus b squared equals the square of the binomial a plus b. Below is a list of the concepts involved in that single sentence. The letters a and b are variables and can be replaced by monomials like or 3 or by polynomials like 3. The square of the binomial a b means (a b)(a b). So, a ab b can be written as the product of two identical factors, a b and a b. Now put these concepts together. The algebraic statement a ab b (a b) means that any trinomial that can be written in the form a ab b can be factored as the square of a binomial using the pattern (a b). When reading a lesson in your book, use these steps. Read the What You ll Learn statements to understand what concepts are being presented. Skim to get a general idea of the content. Take note of any new terms in the lesson by looking for highlighted words. Go back and reread in order to understand all of the ideas presented. Study all of the eamples. Pay special attention to the eplanations for each step in each eample. Read any study tips presented in the margins of the lesson. Reading to Learn. GCF, perfect square trinomial; b c, a b c Turn to page 508 and skim Lesson List three main ideas from Lesson 9-6. Use phrases instead of whole sentences. See margin.. What factoring techniques should be tried when factoring a trinomial? 3. What should you always check for first when trying to factor any polynomial? a greatest common factor 4. Translate the symbolic representation of the Square Root Property presented on page 5 and eplain why it can be applied to problems like (a 4) 49 in Eample 4a. See margin. Answers. () eplains how to factor a perfect square trinomial; () summarizes methods used to factor polynomials; (3) eplains how to solve equations involving perfect squares using the Square Root Property Reading Mathematics The Language of Mathematics For any number n, where n is positive, the square of equals n, then equals plus or minus the square root of n. This property can be applied to the equation (a 4) 49 since the variable a 4 and n 49 in the equation n. Getting Started Before using this page, ask students to think of some common mathematical symbols. As students suggest what they think are some of the more common symbols, write the symbols on the chalkboard. Some eamples might be: the addition sign the subtraction sign the equals sign Teach Interpreting Concepts Write another mathematical sentence on the chalkboard, similar to the one in the tet. Have students interpret the sentence and list the concepts that they have to know in order to interpret the sentence. Learning New Concepts Point out to students that by skimming a new lesson before actually reading it to learn the concepts, they will get an idea about what is important in the lesson. The headings, bold words, and eamples illustrate the important concepts. Assess Study Notebook Ask students to summarize what they have learned about reading and interpreting mathematical concepts in their study notebooks. ELL English Language Learners may benefit from writing key concepts from this activity in their Study Notebooks in their native language and then in English. Reading Mathematics The Language of Mathematics 507

43 Lesson Notes Focus 5-Minute Check Transparency 9-6 Use as a quiz or review of Lesson 9-5. Mathematical Background notes are available for this lesson on p. 47D. can factoring be used to design a pavilion? Ask students: What does the epression 8 represent? The epression represents the side length of the entire square pavilion. What does 44 represent? 44 is the area of the entire square pavilion. How do you know the area of the pavilion is 44 square feet? The square mascot area has an area of 8, or 64 ft because its side length is 8 ft. The problem states that the bricks will cover 80 ft, so the area of the entire structure is 64 80, or 44 ft. What feature of the pavilion tells you that 44 must be a perfect square? The building is square, so the area must be the square of the side length. Since the area is 44, 8, and. Vocabulary perfect square trinomials Study Tip Look Back To review the square of a sum or difference, see Lesson 8-8. Perfect Squares and Factoring Virginia SOL STANDARD A. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. Factor perfect square trinomials. Solve equations involving perfect squares. can factoring be used to design a pavilion? The senior class has decided to build an outdoor pavilion. It will have an 8-foot by 8-foot portrayal of the school s mascot in the center. The class is selling bricks with students names on them to finance the project. If they sell enough bricks to cover 80 square feet and want to arrange the bricks around the art, how wide should the border of bricks be? To solve this problem, you would need to solve the equation (8 ) 44. FACTOR PERFECT SQUARE TRINOMIALS Numbers like 44, 6, and 49 are perfect squares, since each can be epressed as the square of an integer. 44 or or or 7 Products of the form (a b) and (a b), such as (8 ), are also perfect squares. Recall that these are special products that follow specific patterns. (a b) (a b)(a b) (a b) (a b)(a b) a ab ab b a ab ab b a ab b a ab b These patterns can help you factor perfect square trinomials, trinomials that are the square of a binomial. For a trinomial to be factorable as a perfect square, three conditions must be satisfied as illustrated in the eample below. ❶ The first term must be a perfect square. 4 () Squaring a Binomial ❸ The middle term must be twice the product of the square roots of the first and last terms. ()(5) 0 8 ft Factoring a Perfect Square ( 7) ()(7) ()(7) ( 7) (3 4) (3) (3)(4) (3) (3)(4) (3 4) ❷ The last term must be a perfect square ft 508 Chapter 9 Factoring Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 555 Practice, p. 556 Reading to Learn Mathematics, p. 557 Enrichment, p. 558 Assessment, p. 574 Parent and Student Study Guide Workbook, p. 73 Transparencies 5-Minute Check Transparency 9-6 Real-World Transparency 9 Answer Key Transparencies Technology AlgePASS: Tutorial Plus, Lesson 7 Interactive Chalkboard Multimedia Applications

44 Number of Terms Factoring Perfect Square Trinomials Words If a trinomial can be written in the form a ab b or a ab b, then it can be factored as (a b) or as (a b), respectively. Symbols a ab b (a b) and a ab b (a b) Eample () ()(5) (5) or ( 5) Eample Factor Perfect Square Trinomials Determine whether each trinomial is a perfect square trinomial. If so, factor it. a ❶ Is the first term a perfect square? Yes, 6 (4). ❷ Is the last term a perfect square? Yes, ❸ Is the middle term equal to (4)(8)? No, 3 (4)(8) is not a perfect square trinomial. b. 9y y 4 ❶ Is the first term a perfect square? Yes, 9y (3y). ❷ Is the last term a perfect square? Yes, 4. ❸ Is the middle term equal to (3y)()? Yes, y (3y)(). 9y y 4 is a perfect square trinomial. 9y y 4 (3y) (3y)() Write as a ab b. (3y ) Factor using the pattern. In this chapter, you have learned to factor different types of polynomials. The Concept Summary lists these methods and can help you decide when to use a specific method. Factoring Polynomials Factoring Polynomials Factoring Technique Eample or more greatest common factor ( 5) difference of squares perfect square trinomial 3 b c 4 or more a b (a b)(a b) 4 5 ( 5)( 5) a ab b (a b) 6 9 ( 3) a ab b (a b) 4 4 ( ) b c ( m)( n), when m n b and mn c. 9 0 ( 5)( 4) a b c a m n c, a b c when m n b and mn ac. 3( ) ( ) factoring by grouping Then use factoring by grouping. ( )(3 ) a b ay by (a b) y(a b) (a b)( y) 3y 6y 5 0 (3y 6y) (5 0) 3y( ) 5( ) ( )(3y 5) Lesson 9-6 Perfect Squares and Factoring 509 Teach Building on Prior Knowledge In Chapter 8, students learned how to square a binomial such as (a b) or (a b). In this lesson, students will learn how to undo this process to factor perfect square trinomials. FACTOR PERFECT SQUARE TRINOMIALS In-Class Eample Teaching Tip Remind students to look closely at the operation sign in front of the second term of the trinomial. This sign signifies whether the factors are in the form (a b) or (a b). Determine whether each trinomial is a perfect square trinomial. If so, factor it. a yes; (5 3) b. 49y 4y 36 not a perfect square trinomial Assessment During the last lesson of a chapter, it is often good to review some of the major concepts of the chapter to assess whether students have mastered the concepts. The concept summary table on Factoring Polynomials provides a perfect opportunity for review. Briefly review each factoring technique with students. After your review, you might consider giving students a quiz on the different techniques to assess student mastery. New Power Point Lesson 9-6 Perfect Squares and Factoring 509

45 In-Class Eample Factor each polynomial. a ( 4)( 4) In-Class Eample 3 Power Point Teaching Tip Point out that 4 36 is indeed a difference of squares, and can be factored as ( 6)( 6), but remind students that polynomials are not considered to be completely factored if the terms have a GCF greater than. b. 6y 8y 5 (4y 5)(4y 3) SOLVE EQUATIONS WITH PERFECT SQUARES Power Point Teaching Tip Remind students that perfect square trinomials will always have a repeated factor, so they will always have only one solution. Solve Study Tip Alternative Method Note that 4 36 could first be factored as ( 6)( 6). Then the common factor would need to be factored out of each epression. When there is a GCF other than, it is usually easier to factor it out first. Then, check the appropriate factoring methods in the order shown in the table. Continue factoring until you have written the polynomial as the product of a monomial and/or prime polynomial factors. Eample Factor Completely Factor each polynomial. a First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares ( 9) 4 is the GCF. 4( 3 ) and ( 3)( 3) Factor the difference of squares. b This polynomial has three terms that have a GCF of. While the first term is a perfect square, 5 (5), the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is of the form a b c. Are there two numbers m and n whose product is 5 6 or 50 and whose sum is 5? Yes, the product of 5 and 0 is 50 and their sum is m n (5 5) ( 0 6) 5(5 3) (5 3) (5 3)(5 ) Write the pattern. m 5 and n 0 Group terms with common factors. Factor out the GCF from each grouping. 5 3 is the common factor. SOLVE EQUATIONS WITH PERFECT SQUARES When solving equations involving repeated factors, it is only necessary to set one of the repeated factors equal to zero. Eample 3 Solve 0. 4 Solve Equations with Repeated Factors 0 4 Original equation () 0 Recognize as a perfect square trinomial Factor the perfect square trinomial. Set repeated factor equal to zero. Solve for. Thus, the solution set is. Check this solution in the original equation. 50 Chapter 9 Factoring Differentiated Instruction Logical If students do not understand how a second-degree equation can have only one solution, suggest that they graph a perfect square trinomial on a graphing calculator. The graph will immediately reveal how this is possible. The verte of the graph of a perfect square trinomial equation lies on the -ais, hence only one solution. 50 Chapter 9 Factoring

46 Study Tip Reading Math 36 is read as plus or minus the square root of 36. You have solved equations like 36 0 by using factoring. You can also use the definition of square root to solve this equation. Symbols For any number n 0, if n, then n. Eample 9 9 or 3 Use the Square Root Property to Solve Equations Solve each equation. Check your solutions. a. (a 4) 49 (a 4) 49 Original equation a 4 49 Square Root Property a a 4 7 Subtract 4 from each side. a 4 7 or a 4 7 Separate into two equations. 3 Simplify. The solution set is {, 3}. Check each solution in the original equation. b. y 4y 4 5 y 4y 4 5 (y) (y)() 5 Original equation Recognize perfect square trinomial. (y ) 5 Factor perfect square trinomial. y 5 Square Root Property y y 5 Add to each side. y 5 or y 5 Separate into two equations. 7 3 Simplify. Original equation The solution set is { 3, 7}. Check each solution in the original equation. c. ( 3) 5 ( 3) 5 Original equation 3 5 Square Root Property 3 5 Add 3 to each side. Add 36 to each side. Take the square root of each side. Remember that there are two square roots of 36, namely 6 and 6. Therefore, the solution set is { 6, 6}. This is sometimes epressed more compactly as { 6}. This and other eamples suggest the following property. Eample 4 Square Root Property Since 5 is not a perfect square, the solution set is 3 5. Using a calculator, the approimate solutions are 3 5 or about 5.4 and 3 5 or about Lesson 9-6 Perfect Squares and Factoring 5 In-Class Eample Power Point Teaching Tip Point out to students that if they solved 36 0 using factoring, they would also get two solutions. When the factors ( 6)( 6) are set equal to zero, the solutions are {6, 6}. 4 Solve each equation. Check your solutions. a. (b 7) 36 {, 3} b. y y { 6, 4} c. ( 9) 8 { 9 } Lesson 9-6 Perfect Squares and Factoring 5

47 CHECK You can check your answer using a graphing calculator. Graph y ( 3) and y 5. Using the INTERSECT feature of your graphing calculator, find where ( 3) 5. The check of 5.4 as one of the approimate solutions is shown at the right. 3 Practice/Apply Study Notebook Have students complete the definitions/eamples for the remaining terms on their Vocabulary Builder worksheets for Chapter 9. include eplanations on how to factor perfect square trinomials. include any other item(s) that they find helpful in mastering the skills in this lesson. [ 0, 0] scl: by [ 0, 0] scl: Concept Check. See margin.. Determine whether the following statement is sometimes, always, or never true. Eplain your reasoning. never; (a b) a ab b a ab b (a b), b Guided Practice Eercises Eamples 4, , no Factor each polynomial, if possible. If the polynomial cannot be factored, write prime ( 9) 7. c 5c 6 (c 3)(c ) 8. 5a3 80a 5a(a 4)(a 4) ( 7)(4 5) 0. 9g g 4 prime. 3m3 mn m 8n (m )(m )(3m n) Solve each equation. Check your solutions.. 4y 4y 36 0 { 3} 3. 3n 48 { 4} 5. (m 5) 3 冦5 4. a 6a 9 6 {, 7} Odd/Even Assignments Eercises 7 53 are structured so that students practice the same concepts whether they are assigned odd or even problems. Basic: 7 odd, 5 39 odd, odd, 55 56, Average: 7 53 odd, 55 59, Advanced: 8 54 even, 57, 58, Determine whether each trinomial is a perfect square trinomial. If so, factor it. 4. y 8y 6 yes; (y 4) Organization by Objective Factor Perfect Square Trinomials: 7 Solve Equations With Perfect Squares: Assignment Guide 0 3. OPEN ENDED Write a polynomial that requires at least two different factoring techniques to factor it completely. Sample answer: GUIDED PRACTICE KEY About the Eercises. Eplain how to determine whether a trinomial is a perfect square trinomial. Application 苶冧 兹3 6. HISTORY Galileo demonstrated that objects of different weights fall at the same velocity by dropping two objects of different weights from the top of the Leaning Tower of Pisa. A model for the height h in feet of an object dropped from an initial height ho in feet is h 6t ho, where t is the time in seconds after the object is dropped. Use this model to determine approimately how long it took for the objects to hit the ground if Galileo dropped them from a height of 80 feet. about 3.35 s indicates increased difficulty Practice and Apply Determine whether each trinomial is a perfect square trinomial. If so, factor it. For Eercises See Eamples , c 0c 5 7. See margin. 8. a 4a 44. 9n 49 4n 9. 4y 44y. 5a 0ab 44b 3. GEOMETRY The area of a circle is ( ) square inches. What is the diameter of the circle? 8 0 Etra Practice See page 84. positive integer, what is the least possible perimeter measure for the square? Teaching Tip Remind students that any of the factoring methods they have studied thus far can be used in the eercises. 4. GEOMETRY The area of a square is square meters. If is a 5 Chapter 9 Factoring 6 m Answers. Determine if the first term is a perfect square. Then determine if the last term is a perfect square. Finally, check to see if the middle term is equal to twice the product of the square roots of the first and last terms. 5 Chapter 9 Factoring 7. no 8. yes; (a ) 9. yes; (y ) 0. no. yes; (3n 7). yes; (5a b)

48 Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 5. 4k 00 4(k 5)(k 5) (3 4)(3 5) prime 8. 50g 40g 8 (5g ) 9. 9t 3 66t 48t 3t (3t )(t 8) 30. 4a 36b 4(a 3b)(a 3b) 3. 0n 34n 6 (5n )(n 3) 3. 5y 90 5(y 8) (4 3)( 5) 34. 8y 48y 3 (3y 4) g 7g 75 3(3g 5) c 3cd c(45c 3d) 37. (a )(4a 3b ) 37. 4a 3 3a b 8a 6b 38. 5a 7a 6b 4b prime 39. y y z z 40. 4m 4 n 6m 3 n 6m n 4mn (y z )( )( ) mn(m 4n)(m 3) 4. GEOMETRY The volume of a rectangular prism is 3 y 63y 7 9y 3 cubic meters. Find the dimensions of the prism if they can be represented by binomials with integral coefficients. 3y m, 3y m, y 7 m in if 7 4, in if 7 4 Free-Fall Ride Some amusement park free-fall rides can seat 4 passengers across per coach and reach speeds of up to 6 miles per hour. Source: 4. GEOMETRY If the area of the square shown below is square inches, what is the area of the rectangle in terms of? Solve each equation. Check your solutions { 4} 44. 7r 70r 75 {5} a 6 56a y 4y y 3 y a 4 5 a z z 6 { 5, 3} { 4, 4} 5. (y 8) (w 3) p p FORESTRY For Eercises 55 and 56, use the following information. Lumber companies need to be able to estimate the number of board feet that a given log will yield. One of the most commonly used formulas for estimating board feet is L the Doyle Log Rule, B 6 (D 8D 6), where B is the number of board feet, D is the diameter in inches, and L is the length of the log in feet. L 55. Write this formula in factored form. B (D 4) For logs that are 6 feet long, what diameter will yield approimately 56 board feet? 0 in. FREE-FALL RIDE For Eercises 57 and 58, use the following information. The height h in feet of a car above the eit ramp of an amusement park s free-fall ride can be modeled by h 6t s, where t is the time in seconds after the car drops and s is the starting height of the car in feet. 57. How high above the car s eit ramp should the ride s designer start the drop in order for riders to eperience free fall for at least 3 seconds? 44 ft 58. Approimately how long will riders be in free fall if their starting height is 60 feet above the eit ramp? 3.6 s s in. s in. s 3 in. NAME DATE PERIOD 9-6 Enrichment, p. 558 Squaring Numbers: A Shortcut A shortcut helps you to square a positive two-digit number ending in 5. The method is developed using the idea that a two-digit number may be epressed as 0t u. Suppose u 5. (0t 5) (0t 5)(0t 5) 00t 50t 50t 5 00t 00t 5 (0t 5) 00t(t ) 5 s in. Lesson 9-6 Perfect Squares and Factoring 53 In words, this formula says that the square of a two-digit number has t(t ) in the hundreds place. Then is the tens digit and 5 is the units digit. Eample Using the formula for (0t 5), find (8 ) ut: First think Study Study Guide and and Intervention Intervention, p. 553 (shown) and p NAME DATE PERIOD Perfect Squares and Factoring Factor Perfect Square Trinomials Perfect Square Trinomial a trinomial of the form a ab b or a ab b The patterns shown below can be used to factor perfect square trinomials. Squaring a Binomial Factoring a Perfect Square Trinomial (a 4) a (a)(4) 4 a 8a 6 a (a)(4) 4 a 8a 6 (a 4) ( 3) () ()(3) () ()(3) ( 3) Eample Determine whether Eample Factor n 4n 9 is a perfect square Since 5 is not a perfect square, use a different trinomial. If so, factor it. factoring pattern. Since 6n (4n)(4n), the first term is Original trinomial a perfect square. 6 m n 5 Write the pattern. Since 9 3 3, the last term is a m and n 0 perfect square. (6 ) (0 5) Group terms. The middle term is equal to (4n)(3). 4(4 3) 5(4 3) Find the GCF. Therefore, 6n 4n 9 is a perfect (4 5)(4 3) Factor by grouping. square trinomial. Therefore (4 5)(4 3). 6n 4n 9 (4n) (4n)(3) 3 (4n 3) Eercises Determine whether each trinomial is a perfect square trinomial. If so, factor it m 0m 5 3. p 8p 64 yes; ( 8)( 8) yes; (m 5)(m 5) no Factor each polynomial if possible. If the polynomial cannot be factored, write prime y s s (7 0y)(7 0y) ( ) (9 s) 7. 5c 0c r r prime (3 r) (7 )( ) 0. 6m 48m a. b 6b 56 4(m 3) (4 5a)(4 5a) prime a 40ab 5b 5. 8m 3 64m (6 ) (4a 5b) 8m(m 8) Skills Practice, p. 555 and NAME DATE PERIOD 9-6 Practice (Average) Practice, Perfect Squares p. 556 and Factoring (shown) Determine whether each trinomial is a perfect square trinomial. If so, factor it.. m 6m 64. 9s 6s 3. 4y 0y 5 yes; (m 8) yes; (3s ) yes; (y 5) 4. 6p 4p b 4b k 56k 6 yes; (4p 3) no yes; (7k 4) Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 7. 3p q 60q 8 3(p 7)(p 7) ( 7)(3 5) (5q 3) 0. 6t 3 4t t. 6d 8. 30k 38k t(3t )(t 3) 6(d 3) (5k 3)(3k ) 3. 5b 4bc 4. h 60h n 30n 5 3b(5b 8c) 3(h 5) prime 6. 7u 8m 7. w 4 8w c 7cd 8d 7(u m)(u m) (w )(w 3)(w 3) (4c 9d) Solve each equation. Check your solutions. 9. 4k 8k b 0b 0. t 0 7 {} g g 0 3. p p {, } 5. y 8y (h 9) 7. w 6w { 4, } GEOMETRY The area of a circle is given by the formula A r,where r is the radius. If increasing the radius of a circle by inch gives the resulting circle an area of 00 square inches, what is the radius of the original circle? 9 in. 9. PICTURE FRAMING Mikaela placed a frame around a print that measures 0 inches by 0 inches. The area of just the frame itself is 69 square inches. What is the width of the frame?.5 in. Reading to Learn Mathematics, p NAME DATE PERIOD Reading to Learn Mathematics Perfect Squares and Factoring Pre-Activity How can factoring be used to design a pavilion? Read the introduction to Lesson 9-6 at the top of page 508 in your tetbook. On the left side of the equation (8 ) 44, the number 8 in the epression (8 ) represents the length of the pavilion and represents twice the width of the pavilion. On the right side of the equation, the number 44 represents the area of the school mascot in the center of the pavilion, plus the area of the bricks surrounding the center mascot. Reading the Lesson. Three conditions must be met if a trinomial can be factored as a perfect square trinomial.complete the following sentences. The first term of the trinomial 9 6 is (is/is not) a perfect square. The last term of the trinomial, is (is/is not) a perfect square. The middle term is equal to (3)(). The trinomial 9 6 is (is/is not) a perfect square trinomial.. Match each polynomial from the first column with a factoring technique in the second column. Some of the techniques may be used more than once. If none of the techniques can be used to factor the polynomial, write none. a iii i. factor as b c b. 9 4 v ii. factor as a b c c. 5 6 i iii. difference of squares d ii iv. factoring by grouping e. 9y 3y 6 iv v. perfect square trinomial f. 4 4 v vi. factor out the GCF g. 6 vi Helping You Remember 3. Sometimes it is easier to remember a set of instructions if you can state them in a short sentence or phrase. Summarize the conditions that must be met if a trinomial can be factored as a perfect square trinomial. Sample answer: The first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms. 0 0 ELL Lesson 9-6 Perfect Squares and Factoring 53 Lesson 9-6 Lesson 9-6

49 4 Assess Open-Ended Assessment Writing Ask students to look at the concept summary table on page 509, and decide which factoring technique they like best. Then have students write a description of how to use their favorite technique, and why they think it is a better method as compared to one or two of the other methods. Assessment Options Quiz (Lesson 9-6) is available on p. 574 of the Chapter 9 Resource Masters. SOL/EOC Practice Standardized Test Practice 59. HUMAN CANNONBALL A circus acrobat is shot out of a cannon with an initial upward velocity of 64 feet per second. If the acrobat leaves the cannon 6feet above the ground, will he reach a height of 70 feet? If so, how long will it take him to reach that height? Use the model for vertical motion. yes; s CRITICAL THINKING Determine all values of k that make each of the following a perfect square trinomial , k 64 6, k 4, k k k k WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can factoring be used to design a pavilion? Include the following in your answer: an eplanation of how the equation (8 ) 44 models the given situation, and an eplanation of how to solve this equation, listing any properties used, and an interpretation of its solutions. 67. During an eperiment, a ball is dropped off a bridge from a height of 05 feet. The formula 05 6t can be used to approimate the amount of time, in seconds, it takes for the ball to reach the surface of the water of the river below the bridge. Find the time it takes the ball to reach the water to the nearest tenth of a second. C A.3 s B 3.4 s C 3.6 s D.8 s 68. If a ab b a b, then which of the following statements best describes the relationship between a and b? D A a b B a b C a b D a b 6 ft 70 ft Maintain Your Skills Mied Review Solve each equation. Check your solutions. (Lessons 9-4 and 9-5) 69. s m k k w 3w z 7 7z, 7 3 3, 4 5 3, 4 Write the slope-intercept form of an equation that passes through the given point and is perpendicular to the graph of each equation. (Lesson 5-6) (, 4), y y ( 4, 7), y 3 7 y 3 Answer 66. Answers should include the following. The length of each side of the pavilion is 8 or 8 feet. Thus, the area of the pavilion is (8 ) square feet. This area includes the 80 square feet of bricks and the 8 or 64-square foot piece of art, for a total area of 44 square feet. These two representations of the area of the pavilion must be equal, so we can write the equation (8 ) Chapter 9 Factoring 77. NATIONAL LANDMARKS At the Royal Gorge in Colorado, an inclined railway takes visitors down to the Arkansas River. Suppose the slope is 50% or and the vertical drop is 05 feet. What is the horizontal change of the railway? (Lesson 5-) 030 ft Find the net three terms of each arithmetic sequence. (Lesson 4-7) 78. 7, 3, 9, 5, 79. 5, 4.5, 4, 3.5, , 54, 63, 7,, 3, 7 3,.5, 8, 90, 99 (8 ) 44 Original equation 8 Square Root Property 8 or 8 Separate into two equations. 4 0 Solve each equation. 0 Since length cannot be negative, the border should be feet wide. 54 Chapter 9 Factoring

50 Study Guide and Review Vocabulary and Concept Check composite number (p. 474) factored form (p. 475) factoring (p. 48) factoring by grouping (p. 48) greatest common factor (GCF) (p. 476) perfect square trinomials (p. 508) prime factorization (p. 475) prime number (p. 474) prime polynomial (p. 497) Square Root Property (p. 5) Zero Product Property (p. 483) State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence.. The number 7 is an eample of a prime number. false, composite. is the greatest common factor (GCF) of and 4y. true is an eample of a perfect square. false, sample answer: is a factor of 83. true 5. The prime factorization for 48 is 3 4. false, is an eample of a perfect square trinomial. false, difference of squares 7. The number 35 is an eample of a composite number. true is an eample of a prime polynomial. false, sample answer: 9. Epressions with four or more unlike terms can sometimes be factored by grouping. true 0. (b 7)(b 7) is the factorization of a difference of squares. true 9- See pages Eample Factors and Greatest Common Factors Concept Summary The greatest common factor (GCF) of two or more monomials is the product of their common prime factors. Find the GCF of 5 y and 45y. 5 y 3 5 y Factor each number. 45y y y The GCF is 3 5 y or 5y. Circle the common prime factors. Eercises Find the prime factorization of each integer. See Eamples and 3 on page Find the GCF of each set of monomials. See Eample 5 on page , , 8, ab, 4a b 4ab 0. 6mrt, 30m r mr. 0n, 5np 5 5n. 60 y, 35z 3 5 For more information about Foldables, see Teaching Mathematics with Foldables. TM Chapter 9 Study Guide and Review 55 Have students look through the chapter to make sure they have included questions and notes in their Foldables for each lesson of Chapter 9. Encourage students to refer to their Foldables while completing the Study Guide and Review and to use them in preparing for the Chapter Test. Vocabulary and Concept Check This alphabetical list of vocabulary terms in Chapter 9 includes a page reference where each term was introduced. Assessment A vocabulary test/review for Chapter 9 is available on p. 57 of the Chapter 9 Resource Masters. Lesson-by-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabulary PuzzleMaker ELL The Vocabulary PuzzleMaker software improves students mathematics vocabulary using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts (5 questions) Round Skills (4 questions) Round 3 Problem Solving (4 questions) Chapter 9 Study Guide and Review 55

51 Study Guide and Review Chapter 9 Study Guide and Review 9- See pages Factoring Using the Distributive Property Concept Summary Find the greatest common factor and then use the Distributive Property. With four or more terms, try factoring by grouping. Factoring by Grouping: a b ay by (a b) y(a b) (a b)( y) Factoring can be used to solve some equations. Zero Product Property: For any real numbers a and b, if ab 0, then either a 0, b 0, or both a and b equal zero. Eample Factor 3z y 3yz. 3z y 3yz ( 3z) ( y 3yz) ( 3z) y( 3z) Group terms with common factors. Factor out the GCF from each grouping. ( y)( 3z) Factor out the common factor 3z. 5. a(3b 9c 6a) Eercises Factor each polynomial. See Eamples and on pages 48 and y 3( y) 4. 4a b 8ab 6ab(4ab 3) 5. 6ab 8ac 3a 6. a 4ac ab 4bc (a 4c)(a b) 7. 4rs ps mr 6mp 8. 4am 9an 40bm 5bn (r 3p)(s m) (8m 3n)(3a 5b) Solve each equation. Check your solutions. See Eamples and 5 on pages 48 and ( 5) 0 0, (3n 8)(n 6) 0 8 3, , See pages Eample Factoring Trinomials: b c Concept Summary Factoring b c: Find m and n whose sum is b and whose product is c. Then write b c as ( m)( n). Solve a 3a 4 0. Then check the solutions. a 3a 4 0 Original equation 56 Chapter 9 Factoring (a )(a 4) 0 Factor. a 0 or a 4 0 Zero Product Property a a 4 Solve each equation. The solution set is {, 4}. 3. (y 3)(y 4) 33. ( )( 3) Eercises Factor each trinomial. See Eamples 4 on pages 490 and y 7y b 5b 6 (b 6)(b ) r r 36. a 6a m 4mn 3n (r 3)(r 6) (a 0)(a 4) (m 4n)(m 8n) Solve each equation. Check your solutions. See Eample 5 on page y 3y m m 0 { 3, 4} { 5, 8} { 6, } 56 Chapter 9 Factoring

52 Chapter 9 Study Guide and Review Study Guide and Review 9-4 See pages Eample 9-5 See pages Eample Factoring Trinomials: a b c Concept Summary Factoring a b c:find m and n whose product is ac and whose sum is b. Then, write as a m n c and use factoring by grouping. Factor 4. First, factor out the GCF, : 4 (6 7). In the new trinomial, a 6, b and c 7. Since b is positive, m n is positive. Since c is negative, mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 6( 7) or 4, where one factor in each pair is negative. Look for a pair of factors whose sum is. Factors of 4 Sum of Factors, 4 4, 4 4, 9, 9 3, 4 The correct factors are 3 and m n 7 Write the pattern m 3 and n 4 (6 3) (4 7) 3( ) 7( ) ( )(3 7) Group terms with common factors. Factor the GCF from each grouping. is the common factor. Thus, the complete factorization of 4 is ( )(3 7). 4. (m 3)(m 8) 43. (5r )(5r ) Eercises Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. See Eamples 3 on pages 496 and a 9a 3 prime 4. m 3m r 0r z 7z 3 prime 45. b 7b n 6n 45 (4b 3)(3b ) 3(n 5)(n 3) Solve each equation. Check your solutions. See Eample 4 on page r 3r a 3a , 4 5 4, 5, 7 3 Factoring Differences of Squares Concept Summary Difference of Squares: a b (a b)(a b) or (a b)(a b) Sometimes it may be necessary to use more than one factoring technique or to apply a factoring technique more than once. Factor ( 5) The GCF of 3 3 and 75 is 3. 3( 5)( 5) Factor the difference of squares. Chapter 9 Study Guide and Review 57 Chapter 9 Study Guide and Review 57

53 Study Guide and Review Etra Practice, see pages Mied Problem Solving, see page 86. Answers (p. 59). (a )(a ) 3. (y 5)(4m 3p) 4. 5a(3ab a ) 5. (y 3)(3y ) 6. 4(s 5t)(s 5t) 7. ( 4)( 3)( 3) 9-6 See pages Eamples Eercises Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. See Eamples 4 on page y 3 8y 5. 9b 0 prime 5. 4 n 9 r 6 y(y 8)(y 8) Solve each equation by factoring. Check your solutions. See Eample 5 on page b 6 0 { 4, 4} y 0 5 3, a , 9 4 Perfect Squares and Factoring 5. n 3 4 r n 3 4 r Concept Summary If a trinomial can be written in the form a ab b or a ab b, then it can be factored as (a b) or as (a b), respectively. For a trinomial to be factorable as a perfect square, the first term must be a perfect square, the middle term must be twice the product of the square roots of the first and last terms, and the last term must be a perfect square. Square Root Property: For any number n 0, if n, then n. Determine whether 9 4y 6y is a perfect square trinomial. If so, factor it. ❶ Is the first term a perfect square? Yes, 9 (3). ❷ Is the last term a perfect square? Yes, 6y (4y). ❸ Is the middle term equal to (3)(4y)? Yes, 4y (3)(4y). 9 4y 6y (3) (3)(4y) (4y) Write as a ab b. (3 4y) Factor using the pattern. Solve ( 4). ( 4) Original equation 4 Square Root Property 4 4 Add 4 to each side. 4 or 4 Separate into two equations. 5 7 The solution set is { 7, 5}. Eercises Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. See Eample on page a 8a 8 (a 9) 57. 9k k 4 (3k ) r 49r ( 7r) 59. 3n 80n 50 (4n 5) Solve each equation. Check your solutions. See Eamples 3 and 4 on pages 50 and b 3 4b 4b 0 {0, } 6. 49m 6m (c 9) 44 { 3, } b Chapter 9 Factoring 58 Chapter 9 Factoring