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### 5 1factoring out gcf

1. 1. Factoring Out GCF
2. 2. To factor means to rewrite a quantity as a product (in a nontrivial way). Factoring Out GCF
3. 3. To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. Factoring Out GCF
4. 4. To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF
5. 5. Example A. Factor 12 completely. To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF
6. 6. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF
7. 7. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime, incomplete factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF
8. 8. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF A common factor of two or more quantities is a factor belongs to all the quantities.
9. 9. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, A common factor of two or more quantities is a factor belongs to all the quantities.
10. 10. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. A common factor of two or more quantities is a factor belongs to all the quantities.
11. 11. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are A common factor of two or more quantities is a factor belongs to all the quantities.
12. 12. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, A common factor of two or more quantities is a factor belongs to all the quantities.
13. 13. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, A common factor of two or more quantities is a factor belongs to all the quantities.
14. 14. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. A common factor of two or more quantities is a factor belongs to all the quantities.
15. 15. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are A common factor of two or more quantities is a factor belongs to all the quantities.
16. 16. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, A common factor of two or more quantities is a factor belongs to all the quantities.
17. 17. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, A common factor of two or more quantities is a factor belongs to all the quantities.
18. 18. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2, A common factor of two or more quantities is a factor belongs to all the quantities.
19. 19. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, .. A common factor of two or more quantities is a factor belongs to all the quantities.
20. 20. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can’t be written as product besides as 1*x is said to be prime. To factor completely means each factor in the product is prime. Factoring Out GCF Example B. a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factors of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, .. d. The common factor of a(x+y), b(x+y) is (x+y). A common factor of two or more quantities is a factor belongs to all the quantities.
21. 21. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF
22. 22. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36}
23. 23. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12.
24. 24. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a}
25. 25. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a.
26. 26. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2}
27. 27. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2.
28. 28. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} =
29. 29. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4.
30. 30. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. The Extraction Law Distributive law interpreted backward gives the Extraction Law, that is, common factors may be extracted from sums or differences.
31. 31. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. The Extraction Law Distributive law interpreted backward gives the Extraction Law, that is, common factors may be extracted from sums or differences. AB ± AC  A(B±C)
32. 32. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Factoring Out GCF Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. d. GCF{x3y5, x4y6, x5y4} = x3y4. The Extraction Law Distributive law interpreted backward gives the Extraction Law, that is, common factors may be extracted from sums or differences. AB ± AC  A(B±C) This procedure is also called “factoring out a common factor”. To factor, the first step always is to factor out the GCF.
33. 33. Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y
34. 34. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y
35. 35. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4)
36. 36. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a
37. 37. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a
38. 38. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3)
39. 39. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
40. 40. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2
41. 41. (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2
42. 42. (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1)
43. 43. (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
44. 44. Factoring Out GCF We may pull out common factors that are ( )'s. (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
45. 45. Factoring Out GCF We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) – 4(x + y) (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
46. 46. Factoring Out GCF We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) – 4(x + y) Pull out the common factor (x + y) (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
47. 47. Factoring Out GCF We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4) (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
48. 48. Factoring Out GCF We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4) b. Factor (2x – 3)3x – 2(2x – 3) (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
49. 49. Factoring Out GCF We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4) b. Factor (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
50. 50. Factoring Out GCF We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) – 4(x + y) Pull out the common factor (x + y) a(x + y) – 4(x + y) = (x + y)(a – 4) b. Factor (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) = (2x – 3)(3x – 2) (the GCF is y) (the GCF is 2a) (the GCF is 6x2y2) Example D. Factor out the GCF. a. xy – 4y = y(x – 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
51. 51. Factoring Out GCF We sometimes pull out a negative sign from an expression.
52. 52. Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. We sometimes pull out a negative sign from an expression.
53. 53. Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) We sometimes pull out a negative sign from an expression.
54. 54. Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. We sometimes pull out a negative sign from an expression.
55. 55. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. We sometimes pull out a negative sign from an expression.
56. 56. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. We sometimes pull out a negative sign from an expression.
57. 57. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1) Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. We sometimes pull out a negative sign from an expression.
58. 58. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1) b. 3x – 3y + ax – ay Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. We sometimes pull out a negative sign from an expression.
59. 59. We sometimes pull out a negative sign from an expression. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1) b. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice.
60. 60. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1) b. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. We sometimes pull out a negative sign from an expression.
61. 61. Factoring Out GCF Example F. Pull out the negative sign from –2x + 5. –2x + 5 = –(2x – 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x – 5) – 2x + 5 = y(2x – 5) – (2x – 5) = (2x – 5)(y – 1) b. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) We sometimes pull out a negative sign from an expression.
62. 62. Factoring Out GCF Exercise. A. Find the GCF of the listed quantities. Factoring Out GCF 1. {4, 6 } 2. {12, 18 } 3. {32, 20, 12 } 4. {25, 20, 30 } 5. {4x, 6x2 } 6. {12x2y, 18xy2 } 7. {32A2B3, 20A3B3, 12 A2B2} 8. {25x7y6z6, 20y7z5x6, 30z8x7y6 } B. Factor out the GCF. 9. 4 – 6y 10. 12x + 18y 11. 32A + 20B – 12C 12. 25x + 20y – 30 13. –4x + 6x2 14. –12x2y – 18xy2 15. 32A2B3 – 20A3B3 – 12A2B2} 16. 25x7y6z6 – 20y7z5x6 + 30z8x7y6 17. 4x4 – 8x3 + 2x2 18. 20x4 – 5x2 19. x(x – 2) + 3(x – 2) 20. 4x(2x – 3) – 5(2x – 3) C. Factor out the “–”. 21. –2y + 4 22. –3x + 18 23. –5x + 15 24. –8x + 16
63. 63. Factoring Out GCF D. Factor, use grouping if it’s necessary. 25. y2 – 2y + 3y – 6 26. x2 + 3x + 6x + 18 27. y2 – 2y – 3y + 6 28. x2 + 3x – 6x – 18 29. y2 – y + 4y – 4 30. x2 – 5x – 2x + 10 31. 2y2 – y – 6y + 3 32. 3x2 + 2x – 6x – 4 33. 4x2 + 6x – 6x – 9 34. –3x2 + 4x – 6x + 8 35. –5y2 + 10y – 3y + 6 36. –x2 + 3x – 7x + 21 37. 2y2 – xy – 6xy + 3x2 38. 3x2 + 2xy – 6xy – 4y2 39. –5x2 + 2xy – 20xy + 8y2 40. –14x2 + 21xy – 8xy + 12y2