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(1)Factoring Chapter Overview and Pacing. PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. 1. 1. 0.5. 0.5. Factoring Using the Distributive Property (pp. 480–486) Preview: Use algebra tiles and a product mat to factor binomials. • Factor polynomials by using the Distributive Property. • Solve quadratic equations of the form ax 2 bx 0.. 2 (with 9-2 Preview). 1. 1. 0.5. Factoring Trinomials: x2 bx c (pp. 487–494) Preview: Use algebra tiles to factor trinomials. • Factor trinomials of the form x 2 bx c. • Solve equations of the form x 2 bx c 0.. 3 (with 9-3 Preview). 2. 2 (with 9-3 Preview). 1. Factoring Trinomials: ax2 bx c (pp. 495–500) • Factor trinomials of the form ax 2 bx c. • Solve equations of the form ax 2 bx c 0.. 2. 2. 1. 1. Factoring Differences of Squares (pp. 501–506) • Factor binomials that are the differences of squares. • Solve equations involving the differences of squares.. 2. 2. 1. 1. Perfect Squares and Factoring (pp. 508–514) • Factor perfect square trinomials. • Solve equations involving perfect squares.. 2. 2. 1. 1. Study Guide and Practice Test (pp. 515–519) Standardized Test Practice (pp. 520–521). 1. 1. 1. 0.5. Chapter Assessment. 1. 1. 0.5. 0.5. 14. 12. 8. 6. Factors and Greatest Common Factors (pp. 474–479) • Find prime factorizations of integers and monomials. • Find the greatest common factors of integers and monomials.. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 472A Chapter 9 Factoring.

(2) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 523–524. 525–526. 527. 528. 529–530. 531–532. 533. 534. 573. 535–536. 537–538. 539. 540. 573, 575. 541–542. 543–544. 545. 546. 547–548. 549–550. 551. 552. 574. 553–554. 555–556. 557. 558. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 9 RESOURCE MASTERS. 13–14. 13–14. See pages T12–T13.. SC 17. 68. 9-1. 9-1. 69. 9-2. 9-2. 70. 9-3. 9-3. 24, 25. 26. 13–14. GCS 39. 71. 9-4. 9-4. 13–14. GCS 40, SC 18, SM 71–76. 72. 9-5. 9-5. 574. 73. 9-6. 9-6. 559–572, 576–578. 74. Materials. (Preview: algebra tiles, product mat). (Preview: algebra tiles, product mat), graphing calculator. straightedge, scissors, graph paper 27. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet 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 472B.

(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.. 472C. Chapter 9 Factoring. 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 1. The factors of a number can be found by determining all the pairs of numbers whose product is that number. Natural numbers greater than 1 are classified as either prime or composite. Prime numbers have exactly two factors while composite numbers have more than two factors. The number 1 is neither prime nor composite. A prime factorization is the expression of a number as the product of factors that are all prime numbers. The prime factorization of a negative integer is expressed as the product 1 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 exponent greater than 1. So x3 must be written as x x x 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 1 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 1, 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 1 and its additive inverse. Then multiply the GCF of that pair by 1. 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..

(4) Factoring Trinomials: x2 bx c The FOIL method was used to multiply two binomials. Reverse the FOIL method to factor a quadratic polynomial of the form x2 bx 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 (x m) and (x 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 x2 bx 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: ax2 bx c To factor a trinomial in which the coefficient of x2 is not 1, first check to see if the terms of the polynomial have a GCF. If so, factor it out. If the coefficient of x2 is still not 1, or there is no GCF, factor ax2 bx c by making an organized list of the factors of the product of a and c. For example, to factor 8x2 5x 3, make an organized list of the factors of 8 (3), or 24. Look for a pair of factors, m and n, whose sum is equal to b in the trinomial. Then rewrite the trinomial, replacing bx with mx nx. The new polynomial has four terms. Use the factoring by grouping technique shown in Lesson 9-2 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 ax2 bx 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) a2 b2. The binomial a2 b2 is called the difference of two squares and can be factored as the product of a sum and a difference: a2 b2 (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)2 a2 2ab b2, and the square of a difference, (a b)2 a2 2ab b2. These products, a2 2ab b2 and a2 2ab b2, 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 a2 2ab b2 (a b)2 to factor it. If the middle term is negative, use the pattern a2 2ab b2 (a b)2. 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 n. Two solutions result than 0, if x2 n, then x from such equations, one using the positive square root and one using the negative square root.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Solving Equations by Factoring (Lesson 27) Chapter 9 Factoring 472D.

(5) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 473, 479, 486, 494, 500, 506 Practice Quiz 1, p. 486 Practice Quiz 2, p. 500. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 13–14 Quizzes, CRM pp. 573–574 Mid-Chapter Test, CRM p. 575 Study Guide and Intervention, CRM pp. 523–524, 529–530, 535–536, 541–542, 547–548, 553–554. Mixed Review. pp. 479, 486, 494, 500, 506, 514. Cumulative Review, CRM p. 576. Error Analysis. Find the Error, pp. 492, 498, 504 Common Misconceptions, pp. 483, 502. Find the Error, TWE pp. 492, 498, 504 Unlocking Misconceptions, TWE p. 497 Tips for New Teachers, TWE pp. 490, 509. Standardized Test Practice. pp. 479, 486, 494, 500, 503, 505, 514, 519, 520–521. TWE pp. 520–521 Standardized Test Practice, CRM pp. 577–578. Open-Ended Assessment. Writing in Math, pp. 479, 485, 494, 500, 506, 514 Open Ended, pp. 477, 484, 492, 498, 504, 512 Standardized Test, p. 521. Modeling: TWE pp. 479, 506 Speaking: TWE pp. 486, 494 Writing: TWE pp. 500, 514 Open-Ended Assessment, CRM p. 571. Chapter Assessment. Study Guide, pp. 515–518 Practice Test, p. 519. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 559–564 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 565–570 Vocabulary Test/Review, CRM p. 572. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters. Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS. TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 472E. Chapter 9 Factoring.

(6) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson. AlgePASS Lesson. 9-3. 24 Factoring Expressions II. 9-3. 25 Factoring Polynomials Using Algebra Tiles. 9-4. 26 Solving Quadratic Equations Using Factoring. 9-6. 27 Factoring Expressions I. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 68–74 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Glencoe Algebra 1 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, 492, 498, 504, 512) • Reading Mathematics, p. 507 • Writing in Math questions in every lesson, pp. 479, 485, 494, 500, 506, 514 • Reading Study Tip, pp. 489, 511 • WebQuest, p. 479 Teacher Wraparound Edition • Foldables Study Organizer, pp. 473, 515 • Study Notebook suggestions, pp. 477, 480, 484, 488, 492, 498, 504, 507, 512 • Modeling activities, pp. 479, 506 • Speaking activities, pp. 486, 494 • Writing activities, pp. 500, 514 • Differentiated Instruction, (Verbal/Linguistic), p. 475 • ELL Resources, pp. 472, 475, 478, 485, 493, 499, 505, 507, 513, 515 Additional Resources • Vocabulary Builder worksheets require students to define and give examples 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. 527, 533, 539, 545, 551, 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 • Hot Words/Hot Topics Sections 1.2, 3.2, 6.2. For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 9 Factoring 472F.

(7) Notes. Factoring. Have students read over the list of objectives and make a list of any words with which they are not familiar.. • Lesson 9-1 Find the prime factorizations of integers and monomials. • Lesson 9-1 Find the greatest common factors (GCF) for sets of integers and monomials. • Lessons 9-2 through 9-6 Factor polynomials. • Lessons 9-2 through 9-6 Use the Zero Product Property to solve equations.. 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 9-1 9-2 Preview 9-2 9-3 Preview 9-3 9-4 9-5 9-6. NCTM Standards. • • • • •. factored form (p. 475) factoring by grouping (p. 482) prime polynomial (p. 497) difference of squares (p. 501) perfect square trinomials (p. 508). 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 example, the height h in feet of a dolphin that jumps out of the water traveling at 20 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 9-2.. Local Objectives. 1, 2, 6, 8, 9, 10 2, 6, 8, 10 2, 6, 8, 9, 10 2, 6, 8, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 3, 6, 8, 9, 10 2, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 472. Key Vocabulary. Chapter 9 Factoring. 472. 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..

(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 9-2 through 9-6. Distributive Property. Rewrite each expression using the Distributive Property. Then simplify. (For review, see Lesson 1-5.). 1. 3(4 x) 12 3x. 2. a(a 5) a2 5a. 3. 7(n2 3n 1). 7n2. 4. 6y(3y 5y2 y3). 21n 7. For Lessons 9-3 and 9-4. 18y2 30y3 6y4 Multiplying Binomials. Find each product. (For review, see Lesson 8-7.) 5. (x 4)(x 7). 6. (3n 4)(n 5). x2 11x 28. 7. (6a 2b)(9a b). 3n2 11n 20. 8. (x 8y)(2x 12y). 54a2 12ab 2b2. 2x2 4xy 96y2. For Lessons 9-5 and 9-6 10. (3a 2)2. y2 18y 81. 11. (n 5)(n 5). 12. (6p 7q)(6p 7q). n2 25. 36p2 49q2. 9a2 12a 4. For Lesson 9-6. Square Roots. Find each square root. (For review, see Lesson 2-7.) 13. 121 11. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson.. Special Products. Find each product. (For review, see Lesson 8-8.) 9. (y 9)2. 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. 13–14.. 14. 0.0064 0.08. 15.. 25 5 36 6. 16.. 8 2 98 7. For Lesson. Prerequisite Skill. 9-2 9-3 9-4 9-5 9-6. Distributive Property, p. 479 Multiplying Polynomials, p. 486 Factoring by Grouping, p. 494 Square Roots, p. 500 Special Products, p. 506. Make this Foldable to help you organize your notes on 1 factoring. Begin with a sheet of plain 8 " by 11" paper. 2. Fold in Sixths. Fold Again Open. Fold lengthwise, leaving a 12 " tab on the right.. Fold in thirds and then in half along the width.. Cut. Label 9-1. Open. Cut short side along folds to make tabs.. Label each tab as shown.. 9-2 9-3 9-4 9-5 9-6. F a c t o r i n g. As you read and study the chapter, write notes and examples for each lesson under its tab.. Reading and Writing. Chapter 9. Factoring 473. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. 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) Factors and Greatest Common Factors. Lesson Notes. 1 Focus 5-Minute Check Transparency 9-1 Use as a quiz or review of Chapter 8. Mathematical Background notes are available for this lesson on p. 472C. are prime numbers related to the search for extraterrestrial 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.. • Find prime factorizations of integers and monomials. • Find the greatest common factors of integers and monomials.. Vocabulary • • • • •. prime number composite number prime factorization factored form greatest common factor (GCF). are prime numbers related to the search for extraterrestrial life? In the search for extraterrestrial life, scientists listen to radio signals coming from faraway galaxies. 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 18, can be expressed 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 18 square units.. 1 18. 36. 29. The number 18 has 6 factors, 1, 2, 3, 6, 9, and 18. Whole numbers greater than 1 can be classified by their number of factors.. TEACHING TIP An alternative definition is that a prime number is a positive integer with exactly two different factors. You may want to point out that 2 is the only even prime number.. Prime and Composite Numbers Words. Examples. A whole number, greater than 1, whose only factors are 1 and itself, is called a prime number.. 2, 3, 5, 7, 11, 13, 17, 19. A whole number, greater than 1, that has more than two factors is called a composite number.. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18. 0 and 1 are neither prime nor composite.. Study Tip Listing Factors Notice that in Example 1, 6 is listed as a factor of 36 only once.. 474. Example 1 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 36. 1 36 2 18 3 12 49 66 Therefore, the factors of 36, in increasing order, are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Since 36 has more than two factors, it is a composite number.. Chapter 9 Factoring. Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters • Study Guide and Intervention, pp. 523–524 • Skills Practice, p. 525 • Practice, p. 526 • Reading to Learn Mathematics, p. 527 • Enrichment, p. 528. Parent and Student Study Guide Workbook, p. 68 Prerequisite Skills Workbook, pp. 13–14. Transparencies 5-Minute Check Transparency 9-1 Answer Key Transparencies. Technology Interactive Chalkboard.

(10) b. 23. 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.. The only whole numbers that can be multiplied together to get 23 are 1 and 23. Therefore, the factors of 23 are 1 and 23. Since the only factors of 23 are 1 and itself, 23 is a prime number.. When a whole number is expressed as the product of factors that are all prime numbers, the expression is called the prime factorization of the number.. b. 31 The factors are 1 and 31, so the number is prime.. The least prime factor of 90 is 2.. 2 3 15. The least prime factor of 45 is 3.. Teaching Tip. Explain 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 example: 90 3 30 3 3 10 3 3 2 5, which is the same as 2 3 3 5 that was found in Example 2.. 2 3 3 5 The least prime factor of 15 is 3. All of the factors in the last row are prime. Thus, the prime factorization of 90 is 2 3 3 5. Method 2 Use a factor tree. 90 9 10. 90 9 10. 3 3 2 5 9 3 3 and 10 2 5. Study Tip Unique Factorization Theorem The prime factorization of every number is unique except for the order in which the factors are written.. 1 Factor each number. Then. a. 22 The factors are 1, 2, 11, and 22, so the number is composite.. Method 1 90 2 45. Power Point®. classify each number as prime or composite.. Find the prime factorization of 90.. TEACHING TIP. PRIME FACTORIZATION. In-Class Examples. Example 2 Prime Factorization of a Positive Integer. Point out that students could have started with 2 45, 3 30, or 5 18, and found the same prime factorization.. 2 Teach. All of the factors in the last branch of the factor tree are prime. Thus, the prime factorization of 90 is 2 3 3 5 or 2 32 5. Usually the factors are ordered from the least prime factor to the greatest.. 2 Find the prime factorization of 84. 2 2 3 7 or 22 3 7. 3 Find the prime factorization of 132. 1 22 3 11. A negative integer is factored completely when it is expressed as the product of 1 and prime numbers.. Concept Check. Example 3 Prime Factorization of a Negative Integer. Prime Factorization José and Latecia both found the prime factorization of 60. José got 22 3 5, and Latecia got 3 22 5. Explain which is correct. Both are correct. Each number has a unique prime factorization.. Find the prime factorization of 140. 140 . 1 140 1 2 70. 1 2 7 10. Express 140 as 1 times 140. 140 2 70 70 7 10. 1 2 7 2 5 10 2 5 Thus, the prime factorization of 140 is 1 2 2 5 7 or 1 22 5 7.. A monomial is in factored form when it is expressed as the product of prime numbers and variables and no variable has an exponent greater than 1.. www.algebra1.com/extra_examples. Lesson 9-1 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 example. Alternatively, have students write a description of how to factor, using an example as a guide.. Lesson 9-1 Factors and Greatest Common Factors 475.

(11) In-Class Example. Example 4 Prime Factorization of a Monomial. Power Point®. Factor each monomial completely.. 4 Factor each monomial. a. 12a2b3. completely. a.. 12a2b3 2 6 a a b b b. 12 2 6, a2 a a, and b3 b b b. 223aabbb 623. 18x3y3. Thus, 12a2b3 in factored form is 2 2 3 a a b b b.. 2 xxxyyy b. 26rst2 1 2 13 r s t t 32. b. 66pq2 66pq2 1 66 p q q. Teaching Tip. Remind students that prime factorization of a constant can have exponents greater than 1, but prime factorization of variable values cannot.. 1 2 33 p q q. 66 2 33. 1 2 3 11 p q q 33 3 11 Thus, 66pq2 in factored form is 1 2 3 11 p q q.. GREATEST COMMON FACTOR Two or more numbers may have some common prime factors. Consider the prime factorization of 48 and 60.. GREATEST COMMON FACTOR. In-Class Examples. Express 66 as 1 times 66.. 48 2 2 2 2 3 Factor each number. 60 2 2 3 5. Power Point®. Circle the common prime factors.. The integers 48 and 60 have two 2s and one 3 as common prime factors. The product of these common prime factors, 2 2 3 or 12, 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.. 5 Find the GCF of each set of monomials. a. 12 and 18 The GCF is 6 b. 27a2b and 15ab2c The GCF is 3ab. Greatest Common Factor (GCF ) • The GCF of two or more integers is the product of the prime factors common to the integers.. 6 CRAFTS Rene has crocheted 32 squares for an afghan. Each square is 1 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 maximum amount of ribbon she might need to finish the afghan? 66 ft. • 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 1, then the integers or monomials are said to be relatively prime.. Example 5 GCF of a Set of Monomials. Study Tip. Find the GCF of each set of monomials.. Alternative Method. a. 15 and 16. 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 Example 5a. 15: 1, 3, 5, 15 16: 1, 2, 4, 8, 16. Interactive. Chalkboard. The only common factor, and therefore, the greatest common factor, is 1.. 476. 476. Chapter 9 Factoring. Factor each number.. 16 2 2 2 2 Circle the common prime factors, if any. There are no common prime factors, so the GCF of 15 and 16 is 1. This means that 15 and 16 are relatively prime. b. 36x2y and 54xy2z 36x2y 2 2 3 3 x x y. PowerPoint® Presentations. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools. 15 3 5. Factor each number.. 54xy2z 2 3 3 3 x y y z Circle the common prime factors. The GCF of 36x2y and 54xy2z is 2 3 3 x y or 18xy.. Chapter 9 Factoring. Teacher to Teacher Lisa Cook. Kaysville Jr. H.S., Kaysville, UT. “To help in identifying prime factors, I like to have my students explore the Sieve of Eratosthenes. Use a 10-by-10 grid with the numbers 1-100 on it. Cross out 1 since prime numbers are greater than 1. Circle 2 and cross out all multiples of 2. Circle 3 and cross out multiples of 3. Continue with the next odd number until all multiples have been eliminated. The circled numbers are the prime numbers less than 100.".

(12) Example 6 Use Factors GEOMETRY The area of a rectangle is 28 square inches. If the length and width are both whole numbers, what is the maximum perimeter of the rectangle?. 3 Practice/Apply. Find the factors of 28, and draw rectangles with each length and width. Then find each perimeter.. Study Notebook. The factors of 28 are 1, 2, 4, 7, 14, and 28. 28 1. P 1 28 1 28 or 58. 7. 14 4 2. P 4 7 4 7 or 22. P 2 14 2 14 or 32. The greatest perimeter is 58 inches. The rectangle with this perimeter has a length of 28 inches and a width of 1 inch.. Concept Check. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 9. • include explanations 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.. 1. Determine whether the following statement is true or false. If false, provide a counterexample. All prime numbers are odd. false; 2 2. Explain what it means for two numbers to be relatively prime. Their GCF is 1. 3. OPEN ENDED Name two monomials whose GCF is 5x2.. About the Exercises…. Sample answer: 5x2 and 10x3. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4–6 7–9 10–12 13–18 19. 1 2, 3 4 5 6. 12. 1 2 2 5 5 xxxyzz. Find the factors of each number. Then classify each number as prime or composite. 4. 8 1, 2, 4, 8; composite 5. 17 1, 17; prime. 16, 28, 56, 112; composite 9. 150 1 2 3 52. Find the prime factorization of each integer. 7. 45 32 5. 6. 112 1, 2, 4, 7, 8, 14,. 8. 32 1 25. Factor each monomial completely. 10. 4p2 2 2 p p. 11. 39b3c2. 3 13 b b b c c. Odd/Even Assignments Exercises 20–29 and 32–61 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 12. 100x3yz2. Find the GCF of each set of monomials. 14. 18xy, 36y2 18y. 13. 10, 15 5 16. 25n, 21m. 17.. 12a2b,. 90a2b2c. 6a2b. 15. 54, 63, 180 9 18. 15r2, 35s2, 70rs 5. 1 (relatively prime). Application. 19. GARDENING Ashley is planting 120 tomato plants in her garden. In what ways 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? 5 rows of 24 plants,. 6 rows of 20 plants, 8 rows of 15 plants, or 10 rows of 12 plants ★ indicates increased difficulty. Practice and Apply Find the factors of each number. Then classify each number as prime or composite.. 20–27. See margin.. 20. 19. 21. 25. 22. 80. 23. 61. 24. 91. 25. 119. 26. 126. 27. 304. www.algebra1.com/self_check_quiz. Organization by Objective • Prime Factorization: 20–27, 32–39 • Greatest Common Factor: 48–61. Assignment Guide Basic: 21–29 odd, 30, 31, 33–61 odd, 68–86 Average: 21–29 odd, 33–61 odd, 63–66, 68–86 Advanced: 20–28 even, 32–62 even, 63–80 (optional: 81–86). Lesson 9-1 Factors and Greatest Common Factors 477. Answers 20. 1, 19; prime 21. 1, 5, 25; composite 22. 1, 2, 4, 5, 8, 10, 16, 20, 40, 80; composite 23. 1, 61; prime. 24. 1, 7, 13, 91; composite 25. 1, 7, 17, 119; composite 26. 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63 126; composite 27. 1, 2, 4, 8, 16, 19, 38, 76, 152, 304; composite Lesson 9-1 Factors and Greatest Common Factors 477.

(13) NAME ______________________________________________ DATE. Homework Help. ____________ PERIOD _____. Study Guide andIntervention Intervention, 9-1 Study Guide and p. 523 and p.Factors 524 Factors(shown) and Greatest Common. Prime Factorization. When two or more numbers are multiplied, each number is called. a factor of the product. Example. Prime Number. A prime number is a whole number, greater than 1, whose only factors are 1 and itself.. 5. Composite Number. A composite number is a whole number, greater than 1, that has more than two factors.. 10. Prime Factorization. Prime factorization occurs when a whole number is expressed as a product of factors that are all prime numbers.. 45 32 5. Example 1. Example 2. Find the prime factorization of 200. Method 1 200 2 100 2 2 50 2 2 2 25 22255 All the factors in the last row are prime, so the prime factorization of 200 is 23 52.. Factor each number. Then classify each number as prime or composite. a. 28 To find the factors of 28, list all pairs of whole numbers whose product is 28. 1 28 2 14 47 Therefore, the factors of 28 are 1, 2, 4, 7, 14, and 28. Since 28 has more than 2 factors, it is a composite number.. Method 2 Use a factor tree.. b. 31 To find the factors of 31, list all pairs of whole numbers whose product is 31. 1 31 Therefore, the factors of 31 are 1 and 31. Since the only factors of 31 are itself and 1, it is a prime number.. Lesson 9-1. Definition. For Exercises. See Examples. 20–27, 62, 65, 66 32–39 40–47 48–61, 63, 64 28–31, 67. 1. GEOMETRY For Exercises 28 and 29, consider a rectangle whose area is 96 square millimeters and whose length and width are both whole numbers. 28. What is the minimum perimeter of the rectangle? Explain your reasoning. 40 mm 29. What is the maximum perimeter of the rectangle? Explain your reasoning.. 2, 3 4 5. 194 mm 28–29. See margin for explanations. COOKIES For Exercises 30 and 31, use the following information. A bakery packages cookies in two sizes of boxes, one with 18 cookies and the other with 24 cookies. A small number of cookies are to be wrapped in cellophane before they are placed in a box. To save money, the bakery will use the same size cellophane packages for each box.. 6. Extra Practice See page 839.. 30. How many cookies should the bakery place in each cellophane package to maximize the number of cookies in each package? 6 cookies. 200. 31. How many cellophane packages will go in each size box?. 2 100 2 10 10. 3 packages in the box of 18 cookies and 4 packages in the box of 24 cookies.. 22525. All of the factors in each last branch of the factor tree are prime, so the prime factorization of 200 is 23 52.. Find the prime factorization of each integer.. Exercises. 32. 39 3 13. Find the factors of each number. Then classify the number as prime or composite. 1. 41 1, 41; prime. 2. 121 1, 11, 121; composite. 3. 90 1, 2, 3, 5, 6, 9, 10, 15, 18, 30,. 4. 2865 1, 3, 5, 15, 191, 573, 955, 2865;. 45, 90; composite. 36. 115 1 5 23 37. 180. 3. 52. 6. 175 52. 7. 7. 150 1 2 3 . 32x2. 9.. 22222xx. 18m2n. 10.. 233mmn. 40. 66d4. NAME ______________________________________________ DATE. 49a3b2. 77aaabb. 44.. ____________ PERIOD _____. Skills Practice, p. 525 and 9-1 Practice (Average) Practice, p.Greatest 526 Common (shown) Factors and Factors 1. 18 1, 2, 3, 6, 9, 18;. composite 4. 116 1, 2, 4, 29, 58, 116; composite. . 35. 102 2 3 17. 5. 39. 462. 1 2 3 7 11. 45.. 243n3m. 42. 49a3b2. 43. 50gh. 46. 183xyz3. 47. 169a2bc2. 3. 48 1, 2, 3, 4, 6, 8, 12,. 48. 27, 72 9. 49. 18, 35 1. 50. 32, 48 16. 5. 138 1, 2, 3, 6, 23, 46,. 16, 24, 48; composite 6. 211 1, 211; prime. 51. 84, 70 14. 52. 16, 20, 64 4. 53. 42, 63, 105 21. 54. 15a,. 8. 96 1 25 3. 10. 225 32 52. 41. 85x2y2. 128pq2. 69, 138; composite. 7. 52 22 13. 9. 108 22 33. 11. 286 2 11 13. 12. 384 1 27 3. Factor each monomial completely. 13. 30d5. 235ddddd 15. 81b2c3. 22237pqqr. 57.. 21p2q,. 60.. 14m2n2,. 1. 32r2t. 55.. 1. 18mn,. 24d2,. 30c2d. 56. 20gh, 36g2h2 4gh. 6d. 58. 18x, 30xy, 54y 6 2m2n3. 2mn. 61.. 80a2b,. 59. 28a2, 63a3b2, 91b3 7 96a2b3, 128a2b2 16a2b. 1 2 2 2 3 3 m n. 16. 145abc3. 17. 168pq2r. 28b2. 14. 72mn. 3333bbccc. 62. NUMBER THEORY Twin primes are two consecutive odd numbers that are prime. The first pair of twin primes is 3 and 5. List the next five pairs of twin primes. 5, 7; 11, 13; 17, 19; 29, 31; 41, 43. 5 29 a b c c c 18. 121x2yz2. 1 11 11 x x y z z. Find the GCF of each set of monomials. 19. 18, 49 1. 20. 18, 45, 63 9. 21. 16, 24, 48 8. 22. 12, 30, 114 6. 23. 9, 27, 77 1. 24. 24, 72, 108 12. 25. 24fg5, 56f 3g 8fg. 26. 72r2s2, 36rs3 36rs 2. 27. 15a2b, 35ab2 5ab. 28. 28m3n2, 45pq2 1. 29. 40xy2, 56x3y2, 124x2y3 4xy 2. 30. 88c3d, 40c2d2, 32c2d 8c 2d. MARCHING BANDS For Exercises 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.. GEOMETRY For Exercises 31 and 32, use the following information. The area of a rectangle is 84 square inches. Its length and width are both whole numbers. 31. What is the minimum perimeter of the rectangle? 38 in. 32. What is the maximum perimeter of the rectangle? 170 in.. 63. If the bands want to match up in the center of the field, what is the maximum number of rows? 15. RENOVATION For Exercises 33 and 34, use the following information. Ms. Baxter 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 maximum-size square tile Ms. Baxter can use and not have to cut any of the tiles? 12-in. square 34. How many tiles of this size will she need? 12 NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 9-1 Readingto to Learn Learn Mathematics Mathematics, p. 527 Factors and Greatest Common Factors. ELL. How are prime numbers related to the search for extraterrestrial life? Read the introduction to Lesson 9-1 at the top of page 474 in your textbook. If each “beep” counts as one, what are the first two prime numbers?. 2 and 3. 1. Every whole number greater than 1 is either composite or. prime. 64. How many band members will be in each row after the bands are combined? 11. 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 14 to 21 years of age. Source: www.dci.org. Reading the Lesson. NUMBER THEORY For Exercises 65 and 66, use the following information. One way of generating prime numbers is to use the formula 2p 1, where p is a prime number. Primes found using this method are called Mersenne primes. For example, when p 2, 22 1 3. The first Mersenne prime is 3. 65. Find the next two Mersenne primes. 7, 31 ★ 66. Will this formula generate all possible prime numbers? Explain your reasoning. No, it does not generate the first prime number, 2.. Online Research Data Update What is the greatest known prime number? Visit www.algebra1.com/data_update to learn more.. .. 2. 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 except the. 1. 38. 360. 32. 2. 37 1, 37; prime. Find the prime factorization of each integer.. factor. 5. 23. Find the GCF of each set of monomials.. Find the factors of each number. Then classify each number as prime or composite.. Pre-Activity. . 32. Factor each monomial completely. 40–47. See margin.. 52. Factor each monomial completely. 8.. 22. composite. Find the prime factorization of each integer. 5. 600 23. 33. 98 1 2 72 34. 117 32 13. 478. Chapter 9 Factoring. .. 3. Explain why the monomial 5x2y is not in factored form.. The variable x has an exponent that is greater than 1. NAME ______________________________________________ DATE. 4. Explain the steps used below to find the greatest common factor (GCF) of 84 and 120. 84 2 2 3 7 Write the prime factorization of 84.. 9-1 Enrichment Enrichment,. ____________ PERIOD _____. p. 528. 120 2 2 2 3 5 Write the prime factorization of 120.. Finding the GCF by Euclid’s Algorithm. Common prime factors: 2, 2, 3 Identify common prime factors of 84 and 120.. 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 example.. 2 2 3 12 Multiply the common factors to find the GCF of 84 and 120.. Example. Find the GCF of 209 and 532.. Divide the greater number, 532, by the lesser, 209.. 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 expressed as a product of factors. The word prime reminds you that with the possible exception of 1, all of the factors used must be prime numbers.. 478. Chapter 9 Factoring. Divide the remainder into the divisor above. Repeat this process until the remainder is zero. The last nonzero remainder is the GCF.. 2 2095 3 2 418 1 114 2 0 9 114 1 95 1 1 4 95 5 19 9 5 95 0. Answers 28. The factors of 96 whose sum when doubled is the least are 12 and 18. 29. The factors of 96 whose sum when doubled is the greatest are 1 and 96..

(14) ★ 67. GEOMETRY The area of a triangle is 20 square centimeters. What are possible whole-number dimensions for the base and height of the triangle? See margin.. Finding the GCF of distances will help you make a scale model of the solar system. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.. 68a. false; counterexample: a 3, b 4. Standardized Test Practice. 68. CRITICAL THINKING Suppose 6 is a factor of ab, where a and b are natural numbers. Make a valid argument to explain why each assertion is true or provide a counterexample to show that an assertion is false. a. 6 must be a factor of a or of b. b. 3 must be a factor of a or of b. True; see margin for explanation. c. 3 must be a factor of a and of b. false; counterexample: a 3, b 1082 69. WRITING IN MATH. Answer the question that was posed at the beginning of the lesson. See p. 521A.. How are prime numbers related to the search for extraterrestrial life? Include the following in your answer: • a list of the first 30 prime numbers and an explanation of how you found them, and • an explanation of why a signal of this kind might indicate that an extraterrestrial message is to follow. 70. Miko claims that there are at least four ways to design a 120-square-foot rectangular space that can be tiled with 1-foot by 1-foot tiles. Which statement best describes this claim? D A B C D. Her claim is false because 120 is a prime number. Her claim is false because 120 is not a perfect square. Her claim is true because 240 is a multiple of 120. Her claim is true because 120 has at least eight factors.. B. 53. C. 74. 99. D. 117. Maintain Your Skills Mixed Review. Find each product. (Lessons 8-7 and 8-8) 73. 9a2 25 72. (2x . 1)2. 4x2. 4x 1 73. (3a 5)(3a 5). 75. (6r 7)(2r 5). 12r 2 16r 35. 76. (10h k)(2h 5k). 20h2 52hk 5k2. 74. 49p4 56p2 16 74. (7p2 4)(7p2 4) 77. (b 4)(b2 3b 18). b3 7b2 6b 72. Find the value of r so that the line that passes through the given points has the given slope. (Lesson 5-1) 78. (1, 2), (2, r), m 3 7. 3 5. 79. (5, 9), (r, 6), m 0. 80. RETAIL SALES A department store buys clothing at wholesale prices and then marks the clothing up 25% to sell at retail price to customers. If the retail price of a jacket is $79, what was the wholesale price? (Lesson 3-7) $63.20. Getting Ready for the Next Lesson 84.. 12y 2. 24y. PREREQUISITE SKILL Use the Distributive Property to rewrite each expression. (To review the Distributive Property, see Lesson 1-5.) 81. 5(2x 8) 10x 40 82. a(3a 1) 3a2. 83. 2g(3g 4) 6g2 8g. 84. 4y(3y 6). 86. 2x 3x (2 3)x. a 85. 7b 7c 7(b c). 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 9-2 PREREQUISITE SKILL Students will learn to factor polynomials using the Distributive Property in Lesson 9-2. Students should know how to rewrite expressions using the Distributive Property so they can factor polynomials. Use Exercises 81–86 to determine your students’ familiarity with the Distributive Property.. 71. Suppose Ψx is defined as the largest prime factor of x. For which of the following values of x would Ψx have the greatest value? A A. 4 Assess. Lesson 9-1 Factors and Greatest Common Factors 479. Answers 40. 2 3 11 d d d d 41. 5 17 x x y y 42. 7 7 a a a b b 43. 2 5 5 g h 44. 2 2 2 2 2 2 2 p q q 45. 3 3 3 3 3 n n n m 46. 1 3 61 x y z z z 47. 1 13 13 a a b c c 67. base 1 cm, height 40 cm; base 2 cm, height 20 cm; base 4 cm, height 10 cm; base 5 cm, height 8 cm; base 8 cm, height 5 cm; base 10 cm, height 4 cm; base 20 cm, height 2 cm; base 40 cm, height 1 cm 68b.If 6 is a factor of ab, then the prime factorization of ab must contain 2 3 So, 3 must be a factor of either a or b.. Lesson 9-1 Factors and Greatest Common Factors 479.

(15) Algebra Activity. A Preview of Lesson 9-2. A Preview of Lesson 9-2. Getting Started Objective Factor polynomials with algebra tiles. Materials algebra tiles product mat. Factoring Using the Distributive Property 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 1. Use algebra tiles to factor 3x 6.. Model the polynomial 3x 6.. Arrange the tiles into a rectangle. The total area of the rectangle represents the product, and its length and width represent the factors.. Teach • Remind students that they used rectangles at the beginning of Lesson 9-1 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.. x2 1. x. x. 1. 1. x. 3 1. 1. 1. x x x. 1 1 1. The rectangle has a width of 3 and a length of x 2. So, 3x 6 3(x 2).. Activity 2. Use algebra tiles to factor x2 4x.. Model the polynomial x2 4x.. Arrange the tiles into a rectangle.. Assess • In Exercises 1–4, 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 Exercises 5–9, students should recognize that the binomials that cannot be factored can only be modeled in a rectangle with a width of one.. 1 1 1. x4. x. 2. x x x x. x. x. 2. x x x x. The rectangle has a width of x and a length of x 4. So, x2 4x x(x 4).. 9. Binomials can be factored if they can be represented by a rectangle.. Model and Analyze Examples: 2x 2 can be factored and 2x 1 cannot be factored. Use algebra tiles to factor each binomial. 1. 2x 10 2(x 5) 2. 6x 8 2(3x 4). 3. 5x2 2x x(5x 2). 4. 9 3x 3(3 x). Tell whether each binomial can be factored. Justify your answer with a drawing. 5. 4x 10 yes 6. 3x 7 no 7. x2 2x yes 8. 2x2 3 no. 5–8. See pp. 521A–521B for drawings. 9. MAKE A CONJECTURE. Study Notebook You may wish to have students summarize this activity and what they learned from it.. Write a paragraph that explains how you can use algebra tiles to determine whether a binomial can be factored. Include an example of one binomial that can be factored and one that cannot.. 480 Investigating Slope-Intercept Form 480 Chapter 9 Factoring. Resource Manager. 480. Chapter 9 Factoring. Teaching Algebra with Manipulatives. Glencoe Mathematics Classroom Manipulative Kit. • pp. 10–11 (master for algebra tiles) • p. 17 (master for product mat) • p. 156 (student recording sheet). • algebra tiles • product mat.

(16) Factoring Using the Distributive Property • Factor polynomials by using the Distributive Property. • Solve quadratic equations of the form ax2 bx 0.. Vocabulary • factoring • factoring by grouping. Study Tip. 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 151 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 151t 16t2, 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. 2a(6a 8) 2a(6a) 2a(8) 12a2 16a. Look Back To review the Distributive Property, see Lesson 1-5.. You can reverse this process to express a polynomial as the product of a monomial factor and a polynomial factor. 12a2 16a 2a(6a) 2a(8) 2a(6a 8) Thus, a factored form of 12a2 16a is 2a(6a 8). Factoring a polynomial means to find its completely factored form. The expression 2a(6a 8) is not completely factored since 6a 8 can be factored as 2(3a 4).. Example 1 Use the Distributive Property Use the Distributive Property to factor each polynomial. a. 12a2 16a First, find the GCF of 12a2 and 16a. 12a2 2 2 3 a a Factor each number. 16a 2 2 2 2 a Circle the common prime factors. GCF: 2 • 2 • 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. 12a2 16a 4a(3 a) 4a(2 2) Rewrite each term using the GCF. 4a(3a) 4a(4) Simplify remaining factors. 4a(3a 4) Distributive Property Thus, the completely factored form of 12a2 16a is 4a(3a 4).. Lesson Notes. 1 Focus 5-Minute Check Transparency 9-2 Use as a quiz or review of Lesson 9-1. Mathematical Background notes are available for this lesson on p. 472C.. Building on Prior Knowledge In Chapter 1, students were introduced to the Distributive Property and learned how to use it to simplify expressions. 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 151t and 16t2? t • What is the height of the ball when t 0? 0 ft • What is the height of the ball when t 1? 135 ft. Lesson 9-2 Factoring Using the Distributive Property 481. Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters • Study Guide and Intervention, pp. 529–530 • Skills Practice, p. 531 • Practice, p. 532 • Reading to Learn Mathematics, p. 533 • Enrichment, p. 534 • Assessment, p. 573. Parent and Student Study Guide Workbook, p. 69 Prerequisite Skills Workbook, pp. 13–14 School-to-Career Masters, p. 17. Transparencies 5-Minute Check Transparency 9-2 Answer Key Transparencies. Technology Interactive Chalkboard. Lesson x-x Lesson Title 481.

(17) b. 18cd2 12c2d 9cd. 2 Teach. 18cd2 2 3 3 c d d Factor each number. 12c2d 2 2 3 c c d Circle the common prime factors. 9cd 3 3 c d GCF: 3 c d or 3cd. FACTOR BY USING THE DISTRIBUTIVE PROPERTY. In-Class Examples. 18cd2 12c2d 9cd 3cd(6d) 3cd(4c) 3cd(3) Rewrite each term using the GCF. 3cd(6d 4c 3) Distributive Property. Power Point®. Teaching Tip Tell students that one way to find the remaining factors is to divide each term by the GCF.. 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.. 1 Use the Distributive Property to factor each polynomial.. Example 2 Use Grouping. a. 15x 25x2 5x(3 5x). Factor 4ab 8b 3a 6.. b. 12xy 6xy(2 4y 5xy 3) 24xy2. Study Tip. 4ab 8b 3a 6 (4ab 8b) (3a 6) Group terms with common factors. 4b(a 2) 3(a 2) Factor the GCF from each grouping. (a 2)(4b 3) Distributive Property. Factoring by Grouping. CHECK Use the FOIL method.. 30x2y4. Teaching Tip. Remind students that when using the FOIL method, they multiply the First terms, Outer terms, Inner terms, and Last terms.. 2 Factor 2xy 7x 2y 7. (x 1)(2y 7). 3 Factor 15a 3ab 4b 20. (3a 4)(b 5). Sometimes you can group terms in more than one way when factoring a polynomial. For example, the polynomial in Example 2 could have been factored in the following way. 4ab 8b 3a 6 (4ab 3a) (8b 6) a(4b 3) 2(4b 3) (4b 3)(a 2) Notice that this result is the same as in Example 2.. Concept Check Factoring Using the Distributive Property Malcolm. and Fatima each factored the polynomial 2ax 6cx ab 3bc. Malcolm’s answer was (2x b)(a 3c) and Fatima’s was (a 3c)(2x b). Which is correct? Explain your answer. Both are correct. The order in which factors are multiplied does not affect the product.. F. O. I. L. (a 2)(4b 3) (a)(4b) (a)(3) (2)(4b) (2)(3) 4ab 3a. . . 8b. Recognizing binomials that are additive inverses is often helpful when factoring by grouping. For example, 7 y and y 7 are additive inverses because their sum is 0. By rewriting 7 y in the factored form 1(y 7), factoring by grouping is made possible in the following example.. Example 3 Use the Additive Inverse Property Factor 35x 5xy 3y 21.. Study Tip Factoring Trinomials Since the order in which factors are multiplied does not affect the product, (5x 3)(y 7) is also a correct factoring of 35x 5xy 3y 21.. 35x 5xy 3y 21 (35x 5xy) (3y 21) 5x(7 y) 3(y 7) 5x(1)(y 7) 3(y 7) 5x(y 7) 3(y 7) (y 7)(5x 3). Group terms with common factors. Factor the GCF from each grouping. 7 y 1(y 7) 5x(1) 5x Distributive Property. Factoring by Grouping • Words. A polynomial can be factored by grouping if all of the following situations exist. • 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 ax bx ay by x(a b) y(a b) (a b)(x y). 482. 482. Chapter 9 Factoring. 6 . Chapter 9 Factoring.

(18) SOLVE EQUATIONS BY FACTORING Some equations can be solved by factoring. Consider the following products. 6(0) 0. 0(3) 0. (5 5)(0) 0. 2(3 3) 0. Notice that in each case, at least one of the factors is zero. These examples 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.. Example 4 Solve an Equation in Factored Form. SOLVE EQUATIONS BY FACTORING. In-Class Examples. 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.. 4 Solve (x 2)(4x 1) 0.. 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.. Then check the solutions. 2, 14 . (d 5)(3d 4) 0 d 5 0 or d5. Teaching Tip. Original equation. 3d 4 0 3d 4. Remind students that for the Zero Product Property to work, one of two factors is equal to zero. Therefore, students must factor x2 7x before they can assume that one term is equal to zero.. Set each factor equal to zero. Solve each equation.. 4 d 3. The solution set is 5, . 4 3. 4 3. CHECK Substitute 5 and for d in the original equation. (d 5)(3d 4) 0. (d 5)(3d 4) 0. (5 5)[3(5) 4] ⱨ 0. 5 3 4 ⱨ 0. (0)(19) ⱨ 0. (0) ⱨ 0. 4 3. 5 Solve 4y 12y2. Then check 1 the solutions. 0, 3. 4 3. 19 3. 00 . 00 . 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 Example 5 by dividing each side of the equation by x. Remember, however, that x is an unknown quantity. If you divide by x, you may actually be dividing by zero, which is undefined.. Example 5 Solve an Equation by Factoring Solve x2 7x. Then check the solutions. Write the equation so that it is of the form ab 0. x2 7x Original equation x2 7x 0 Subtract 7x from each side. x(x 7) 0 Factor the GCF of x2 and 7x, which is x. x 0 or x 7 0 Zero Product Property x 7 Solve each equation. The solution set is {0, 7}. Check by substituting 0 and 7 for x in the original equation.. www.algebra1.com/extra_examples. Lesson 9-2 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 9-2 Factoring Using the Distributive Property 483.

(19) 3 Practice/Apply Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 9. • include explanations 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 2. an equation that can be written as a product of factors that equal 0. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4–9 10–15. 1–3 4, 5. 1. Write 4x2 12x as a product of factors in three different ways. Then decide which of the three is the completely factored form. Explain your reasoning. See margin. 2. OPEN ENDED Give an example of the type of equation that can be solved by using the Zero Product Property. 3. Explain why (x 2)(x 4) 0 cannot be solved by dividing each side by x 2.. The division would eliminate 2 as a solution.. Factor each polynomial. 7. 2ab(a2b 4 8ab2) 4. 9x2 36x 9x(x 4). 5. 16xz 40xz2 8xz(2 5z). 6. 24m2np2 36m2n2p 12m2np(2p 3n) 7. 2a3b2 8ab 16a2b3 8. 5y2 15y 4y 12 (5y 4)(y 3). 9. 5c 10c2 2d 4cd. (5c 2d)(1 2c). Solve each equation. Check your solutions. 10. h(h 5) 0 {0, 5}. Application. 11. (n 4)(n 2) 0. {2, 4}. . 5 12. 5m 3m2 0, 3. PHYSICAL SCIENCE For Exercises 13–15, 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 100t 16t2, where t is the time in seconds after the flare is launched. 13. At what height is the flare when it returns to the sea? 0 ft. h 100t 16t 2. 14. Let h 0 in the equation h 100t 16t2 and solve for t. 0, 6.25 15. How many seconds will it take for the flare to return to the sea? Explain your reasoning.. About the Exercises… Organization by Objective • Factor by Using the Distributive Property: 16–39 • Solve Equations by Factoring: 48–59 Odd/Even Assignments Exercises 16–39 and 44–59 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 100 ft/s. 6.25 s; The answer 0 is not reasonable since it represents the time at which the flare is launched.. h0. ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 16–29, 40–47 30–39 48–61. 1 2, 3 4, 5. Extra Practice See page 840.. Assignment Guide Basic: 17–39 odd, 40–43, 47–59 odd, 62–81 Average: 17–39 odd, 42, 43, 45, 47–61 odd, 62–81 Advanced: 16–38 even, 44, 45, 46–60 even, 62–75 (optional: 76–81) All: Practice Quiz 1 (1–10). Factor each polynomial. 16–39. See p. 521A. 16. 5x 30y. 17. 16a 4b. 18. a5b a. 19. x3y2 x. 20. 21cd 3d. 21. 14gh 18h. 22. 15a2y 30ay. 23. 8bc2 24bc. 24. 12x2y2z 40xy3z2. 25. 18a2bc2 48abc3. 26. a a2b2 a3b3. 27. 15x2y2 25xy x. . 30. x2 2x 3x 6. 28.. 12ax3. 31.. x2. . 20bx2. 32cx. 5x 7x 35. 29.. 3p3q. 32.. 4x2 . 9pq2. 36pq. 14x 6x 21. 33. 12y2 9y 8y 6. 34. 6a2 15a 8a 20. 35. 18x2 30x 3x 5. 36. 4ax 3ay 4bx 3by. 37. 2my 7x 7m 2xy. 38. 8ax 6x 12a 9. 39. 10x2 14xy 15x 21y. GEOMETRY For Exercises 40 and 41, use the following information. A quadrilateral has 4 sides and 2 diagonals. A pentagon has 5 sides and 5 diagonals. 1. 3. You can use n2 n to find the number of diagonals in a polygon with n sides. 2 2 1 40. Write this expression in factored form. n(n 3) 2 41. Find the number of diagonals in a decagon (10-sided polygon). 35 484. Chapter 9 Factoring. Answers 1. Sample answers: 4(x 2 3x), x(4x 12), or 4x(x 3); 4x(x 3); 4x is the GCF of 4x 2 and 12x.. 484. Chapter 9 Factoring. . 63. Answers should include the following. • Let h 0 in the equation h 151t 16t 2. To solve 0 151t 16t 2, factor the right-hand side as t(151 16t). Then, since t(151 16t) 0, either t 0 or 151 16t 0. Solving each equation for t, we find that t 0 or t 9.44. • 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) SOFTBALL For Exercises 42 and 43, use the following information. Albertina is scheduling the games for a softball league. To find the number of 1 2. 63 games GEOMETRY Write an expression in factored form for the area of each shaded region.. ★ 45.. 2 2. b. r. p. 529 (shown) and p.Property 530 Factoring Using the Distributive. Factor by Using the Distributive Property. represents the number of games needed for each team to play each other team exactly once and n represents the number of teams. 1 42. Write this equation in factored form. g n(n 1) 2 43. How many games are needed for 7 teams to play each other exactly 3 times?. 44.. r. The Distributive Property has been used to multiply a polynomial by a monomial. It can also be used to express 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). x(y z ) xy xz. xy xz x(y z). 6y (2x 1) 6y (2x) 6y(1) 12xy 6y. 12xy 6y 6y (2x) 6y(1) 6y(2x 1). Example 1 Use the Distributive Property to factor 12mn 80m2.. Example 2 Factor 6ax 3ay 2bx by by grouping.. Find the GCF of 12mn and 80m2. 12mn 2 2 3 m n 80m2 2 2 2 2 5 m m GCF 2 2 m or 4m. 6ax 3ay 2bx by (6ax 3ay) (2bx by) 3a(2x y) b(2x y) (3a b)(2x y). 12mn 80m2 4m(3 n) 4m(2 2 5 m) 4m(3n) 4m(20m) 4m(3n 20m) Thus 12mn 80m2 4m(3n 20m).. Exercises. 2. a. Factor each polynomial.. 2r 2(4. 2. ). 1. 24x 48y. 2. 30mn2 m2n 6n. 3. q4 18q3 22q. 4. 9x2 3x. 5. 4m 6n 8mn. 6. 45s3 15s2. 7. 14c3 42c5 49c4. 8. 55p2 11p4 44p5. 9. 14y3 28y2 y. n(30mn m 2 6). 3x(3x 1). y(14y 2 28y 1). 12. 6y 12x 8z. GEOMETRY Find an expression for the area of a square with the given perimeter.. 13. x2 2x x 2. 14. 6y2 4y 3y 2. 15. 4m2 4mn 3mn 3n2. 46. P 12x 20y in.. 16. 12ax 3xz 4ay yz. 17. 12a2 3a 8a 2. 18. xa ya x y. 47. P 36a 16b cm. 81a2 72ab 16b2 cm2. . . (x y)(a 1) ____________ PERIOD _____. 1. 64 40ab. 2. 4d2 16. 3. 6r2s 3rs2. 4. 15cd 30c2d 2. 5. 32a2 24b2. 6. 36xy2 48x2y. 7. 30x3y 35x2y2. 8. 9c3d2 6cd3. 9. 75b2c3 60bc3. 8(8 5ab). . . . (4m 3n)(m n). (4a 1)(3a 2). Factor each polynomial.. . . (2y 1)(3y 2). p. 531 and Practice, p. 532 (shown) Factoring Using the Distributive Property. 51. (3y 9)(y 7) 0 {3, 7} 5 7 53. (4n 5)(3n 7) 0 , 4 3 2 55. 7d 35d 0 {0, 5} 6 57. 7x2 6x 0, 7 59. 20x2 15x 3 , 0 4. . 2(3y 6x 4z). Skills Practice, 9-2 Practice (Average). 50. (q 4)(3q 15) 0 {4, 5} 3 8 52. (2b 3)(3b 8) 0 , 2 3 2 54. 3z 12z 0 {4, 0} 5 56. 2x2 5x 0, 2 58. 6x2 4x 2 , 0 3. . ab(4a 28b 7). NAME ______________________________________________ DATE. 49. a(a 16) 0 {16, 0}. . (x 1)(x 2) (3x y)(4a z). 48. x(x 24) 0 {0, 24}. . . 4(d 2 4). 15cd(1 2cd) 5x 2y(6x 7y). 12xy(3y 4x). 3cd 2(3c 2 2d). 15bc 3(5b 4). 11. 5x3y2 10x2y 25x. 12. 9ax3 18bx2 24cx. 13. x2 4x 2x 8. 14. 2a2 3a 6a 9. 15. 4b2 12b 2b 6. 16. 6xy 8x 15y 20. 17. 6mn 4m 18n 12. 18. 12a2 15ab 16a 20b. 8pq(pq 3q 2 2) (x 2)(x 4). (2x 5)(3y 4). 5x(x 2y 2 2xy 5). 3x(3ax 2 6bx 8c). (a 3)(2a 3). (4b 2)(b 3). (2m 6)(3n 2). (3a 4)(4a 5b). Solve each equation. Check your solutions. 20. 4b(b 4) 0. {0, 32} 22. (a 6)(3a 7) 0. . 7 6, 3. . . 24. (4y 8)(3y 4) 0. . 2, 43 . 26. 8p2 4p 0. 27. 9x2 27x. 0, 21 . {10, 0}. . {2, 3}. 23. (2y 5)( y 4) 0. 5 , 4 2. 25. 2z2 20z 0. 5 0, 6. 21. ( y 3)( y 2) 0. {4, 0}. 28. 18x2 15x. 61. 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 2 48t 16t2. How long was the ball in the air if the catcher catches the ball when it is 2 feet above the ground? about 2.8 s. 3rs(2r s). 8(4a 2 3b2). 10. 8p2q2 24pq3 16pq. 19. x(x 32) 0. 60. MARINE BIOLOGY In a pool at a water park, a dolphin jumps out of the water traveling at 20 feet per second. Its height h, in feet, above the water after t seconds is given by the formula h 20t 16t2. How long is the dolphin in the air before returning to the water? 1.25 s. {0, 3}. 29. 14x2 21x. . . 30. 8x2 26x. . 13 , 0 4. 3 , 0 2. LANDSCAPING For Exercises 31 and 32, 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 1 represented by the equation A b2 b. 2. 12. . 31. Write this equation in factored form. A b b 1. 32. Suppose the base of the flower bed is 16 feet. What will be its area? 112 ft2. 62. CRITICAL THINKING Factor ax y axby aybx bx y. (ax bx)(ay by). 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 16t2. How long was the rocket in the air before it returned to the ground? 3.75 s NAME ______________________________________________ DATE. 63. WRITING IN MATH. Answer the question that was posed at the beginning of the lesson. See margin.. ____________ PERIOD _____. Reading 9-2 Readingto to Learn Learn Mathematics. ELL Mathematics, p. 533 Factoring Using the Distributive Property. Pre-Activity. How can you determine how long a baseball will remain in the air?. How can you determine how long a baseball will remain in the air?. Read the introduction to Lesson 9-2 at the top of page 481 in your textbook.. Include the following in your answer: • an explanation 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 context of the problem.. the velocity of the baseball in feet per second at the instant it is thrown. www.algebra1.com/self_check_quiz. In the formula h 151t 16t2, what does the number 151 represent?. Reading the Lesson 1. Factoring a polynomial means to find its completely factored form. a. The expression x(6x 9) is a factored form of the polynomial 6x2 9x. 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 expression.. Lesson 9-2 Factoring Using the Distributive Property 485 b. Provide an example of a completely factored polynomial.. Sample answer: 2b(5b 4) is the completely factored form of 10b 2 8b. NAME ______________________________________________ DATE. 9-2 Enrichment Enrichment,. ____________ PERIOD _____. Perfect, Excessive, Defective, and Amicable Numbers A perfect number is the sum of all of its factors except itself. Here is an example: 28 1 2 4 7 14 There are very few perfect numbers. Most numbers are either excessive or defective. An excessive number is greater than the sum of all of its factors except itself. A defective number is less than this sum. Two numbers are amicable if the sum of the factors of the first number, except for the number itself, equals the second number, and vice versa.. c. Provide an example of a polynomial that is not completely factored.. Sample answer: b(10b 8) is a factored form of 10b 2 8b, but is not the completely factored form.. p. 534. 2. The polynomial 5ab 5b2 3a 6b can be rewritten as 5b(a b) 3(a 2b). Does this indicate that the original polynomial can be factored by grouping? Explain.. No, because 5b(a b) and 3(a 2b) do not have a common factor other than 1. 3. The polynomial 3x2 3xy 2x 2y can be rewritten as 3x(x y) 2(x y). Does this indicate that the original polynomial can be factored by grouping? Explain.. Yes, because 3x(x y) and 2(x y) have the common factor (x y).. Helping You Remember 4. How would you explain 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 expressions.. Solve each problem. 1. Write the perfect numbers between 0 and 31.. 6, 28 2 W it th. i. b. b t. 0. d 31. Lesson 9-2 Factoring Using the Distributive Property 485. Lesson 9-2. Source: National Sea Grant Library. 11p 2(5 p 2 4p 3). 11. 4a2b 28ab2 7ab. 4x(1 3x 4x 2). Solve each equation. Check your solutions.. Online Research. 15s 2(3s 1). 10. 4x 12x2 16x3. 9x2 30xy 25y2 in2. For information about a career as a marine biologist, visit: www.algebra1.com/ careers. q(q 3 18q 2 22). 2(2m 3n 4mn). 7c 3(2 6c 2 7c). 4(a b 4). Marine biologists study factors that affect organisms living in and near the ocean.. Check using the FOIL method. (3a b)(2x y) 3a(2x) (3a)( y) (b)(2x) (b)( y) 6ax 3ay 2bx by ✓. Write each term as the product of the GCF and its remaining factors.. 24(x 2y). Marine Biologist. ____________ PERIOD _____. Lesson 9-2. 1 2. games she needs to schedule, she uses the equation g n2 n, where g. NAME ______________________________________________ DATE. Study Guide andIntervention Intervention, 9-2 Study Guide and.

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