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Splash Screen. Contents. Lesson 9-1 Factors and Greatest Common Factors Lesson 9-2 Factoring Using the Distributive Property Lesson 9-3 Factoring Trinomials: x 2 + bx + c Lesson 9-4 Factoring Trinomials: ax 2 + bx + c Lesson 9-5 Factoring Differences of Squares

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  1. Splash Screen

  2. Contents Lesson 9-1Factors and Greatest Common Factors Lesson 9-2Factoring Using the Distributive Property Lesson 9-3Factoring Trinomials: x2+ bx + c Lesson 9-4Factoring Trinomials: ax2 + bx + c Lesson 9-5Factoring Differences of Squares Lesson 9-6Perfect Squares and Factoring

  3. Lesson 1 Contents Example 1Classify Numbers as Prime or Composite Example 2Prime Factorization of a Positive Integer Example 3Prime Factorization of a Negative Integer Example 4Prime Factorization of a Monomial Example 5GCF of a Set of Monomials Example 6Use Factors

  4. Example 1-1a Factor 22. Then classify it as prime or composite. To find the factors of 22, list all pairs of whole numbers whose product is 22. Answer: Since 22 has more than two factors, it is a compositenumber. The factors of 22, in increasing order, are 1, 2, 11, and 22.

  5. Example 1-1a Factor 31. Then classify it as prime or composite. The only wholenumbers that can be multiplied together to get 31 are 1 and 31. Answer: The factors of 31 are 1 and 31.Since the only factors of 31 are 1 and itself, 31 is a prime number.

  6. Example 1-1b Factor each number. Then classify it as primeor composite. a. 17 b. 25 Answer: 1, 17; prime Answer: 1, 5, 25; composite

  7. The least prime factor of 84 is 2. The least prime factor of 42 is 2. The least prime factor of 21 is 3. Answer: Thus, the prime factorization of 84 is Example 1-2a Find the prime factorization of 84. Method 1 All of the factors in the last row are prime.

  8. 21 4 3 7 2 2 and Answer: Thus, the prime factorization of 84 is or Example 1-2a Method 2 Use a factor tree. 84 All of the factors in the last branch of the factor tree are prime.

  9. Answer: or Example 1-2b Find the prime factorization of 60.

  10. Express –132 as –1 times 132. / \ / \ / \ Answer: The prime factorization of –132 is or Example 1-3a Find the prime factorization of –132.

  11. Answer: Example 1-3b Find the prime factorization of –154.

  12. Factor completely. Answer: in factored form is Example 1-4a

  13. Factor completely. Express –26 as –1 times 26. Answer: in factored form is Example 1-4a

  14. Factor each monomial completely. a. b. Answer: Answer: Example 1-4b

  15. Factor each number. The integers 12 and 18 have one 2 and one 3 as common prime factors. The product of thesecommon prime factors, or 6, is the GCF. Example 1-5a Find the GCF of 12 and 18. Circle the common prime factors. Answer: The GCF of 12 and 18 is 6.

  16. Find the GCF of . Factor each number. Answer: The GCF of and is . Example 1-5a Circle the common prime factors.

  17. Find the GCF of each set of monomials. a. 15 and 35 b. and Answer: Example 1-5b Answer: 5

  18. Example 1-6a CraftsRene 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 an afghan? Find the factors of 32 and draw rectangles with each length and width. Then find each perimeter. The factors of 32 are 1, 2, 4, 8, 16, 32.

  19. Example 1-6a The greatest perimeter is 66 feet. The afghan with this perimeter has a length of 32 feet and a width of 1 foot. Answer: The maximum amount of ribbon Rene will need is 66 feet.

  20. Example 1-6b Mary wants to plant a rectangular flower bed in her front yard with a stone border. The area of the flower bed will be 45 square feet and the stones are one foot square each. What is the maximum number of stones that Mary will need to go around all four sides of the flower bed? Answer: 92 feet

  21. End of Lesson 1

  22. Lesson 2 Contents Example 1Use the Distributive Property Example 2Use Grouping Example 3Use the Additive Inverse Property Example 4Solve an Equation in Factored Form Example 5Solve an Equation by Factoring

  23. Use the Distributive Property to factor . First, find the CGF of 15x and . Factor each number. GFC: Rewrite each term using the GCF. Simplify remaining factors. Distributive Property Example 2-1a Circle the common prime factors. Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF.

  24. Answer: The completely factored form of is Example 2-1a

  25. Use the Distributive Property to factor . Factor each number. GFC: or Rewrite each term using the GCF. Distributive Property Answer: The factored form of is Example 2-1a Circle the common prime factors.

  26. Use the Distributive Property to factor each polynomial. a. b. Answer: Answer: Example 2-1b

  27. Factor Group terms with common factors. Factor the GCFfrom each grouping. Answer: Distributive Property Example 2-2a

  28. Factor Answer: Example 2-2b

  29. Factor Group terms with common factors. Factor GCF from each grouping. Answer: Distributive Property Example 2-3a

  30. Factor Answer: Example 2-3b

  31. Solve Then check the solutions. If , then according to the Zero Product Property either or Original equation or Set each factor equal to zero. Solve each equation. Answer: The solution set is Example 2-4a

  32. Check Substitute 2 and for x in the original equation. Example 2-4a

  33. Solve Then check the solutions. Example 2-4b Answer: {3, –2}

  34. Solve Then check the solutions. Write the equation so that it is of the form Original equation Subtract from each side. Factor the GCF of 4y andwhich is 4y. Zero Product Property or Solve each equation. Example 2-5a

  35. Answer: The solution set is Check by substituting 0 and for y in the original equation. Example 2-5a

  36. Solve Answer: Example 2-5b

  37. End of Lesson 2

  38. Lesson 3 Contents Example 1b and c Are Positive Example 2b Is Negative and c Is Positive Example 3b Is Positive and c Is Negative Example 4b Is Negative and c Is Negative Example 5Solve an Equation by Factoring Example 6Solve a Real-World Problem by Factoring

  39. Factor In this trinomial, and You need to find the two numbers whose sum is 7 and whose product is 12. Make an organized list of the factors of 12, and look for the pair of factors whose sum is 7. Write the pattern. and Answer: Example 3-1a The correct factors are 3 and 4.

  40. F O I L FOIL method Simplify. Example 3-1a Check You can check the result by multiplying the two factors.

  41. Factor Answer: Example 3-1b

  42. Factor In this trinomial, and This means is negative and mn is positive. So m and n must both be negative. Therefore,make a list of the negative factors of 27, and look for the pair whose sum is –12. Write the pattern. Answer: and Example 3-2a The correct factors are –3 and –9.

  43. Check You can check this result by using a graphing calculator. Graph andon the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly. Example 3-2a

  44. Factor Answer: Example 3-2b

  45. Factor In this trinomial, and Thismeans is positive and mn is negative, so either m or n is negative, but not both. Therefore,make a list of the factors of –18 where one factorof each pair is negative. Look for the pair of factors whose sum is 3. Example 3-3a The correct factors are –3 and 6.

  46. Write the pattern. and Answer: Example 3-3a

  47. Factor Answer: Example 3-3b

  48. Factor Since and is negative and mn is negative. So either m or n is negative, but not both. Example 3-4a The correct factors are 4 and –5.

  49. Write the pattern. and Answer: Example 3-4a

  50. Factor Answer: Example 3-4b

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