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# 5 1factoring out gcf

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• 1. Factoring Out GCF
• 2. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way).
• 3. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;t be written as product besides as 1*x is said to be prime.
• 4. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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.
• 5. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely.
• 6. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2
• 7. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime, incomplete factored completely
• 8. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities.
• 9. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5,
• 10. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
• 11. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are
• 12. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2,
• 13. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a,
• 14. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a, 2a.
• 15. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are
• 16. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3,
• 17. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x,
• 18. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2,
• 19. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor of 4ab, 6a are 2, a, 2a. c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..
• 20. Factoring Out GCF To factor means to rewrite a quantity as a product (in a nontrivial way). A quantity x that can&#x2019;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. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 not prime factored completely A common factor of two or more quantities is a factor belongs to all the quantities. Example B. a. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor. b. The common factor 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).
• 21. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors.
• 22. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example C. Find the GCF of the given quantities. a. GCF{24, 36}
• 23. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12.
• 24. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a}
• 25. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Example C. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a.
• 26. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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 &#xB1; AC &#xF0E0; A(B&#xB1;C)
• 32. Factoring Out GCF The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. 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 &#xB1; AC &#xF0E0; A(B&#xB1;C) This procedure is also called &#x201C;factoring out a common factor&#x201D;. To factor, the first step always is to factor out the GCF.
• 33. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y
• 34. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y
• 35. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4)
• 36. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) b. 4ab + 6a
• 37. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) b. 4ab + 6a
• 38. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) b. 4ab + 6a = 2a(2b) + 2a(3)
• 39. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
• 40. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2
• 41. Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) Factoring Out GCF c. 12x2y3 + 6x2y2 (the GCF is 6x2y2)
• 42. Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) Factoring Out GCF c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) (the GCF is 6x2y2)
• 43. Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) Factoring Out GCF c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2)
• 44. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s.
• 45. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) &#x2013; 4(x + y)
• 46. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) &#x2013; 4(x + y) Pull out the common factor (x + y)
• 47. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) &#x2013; 4(x + y) Pull out the common factor (x + y) a(x + y) &#x2013; 4(x + y) = (x + y)(a &#x2013; 4)
• 48. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) &#x2013; 4(x + y) Pull out the common factor (x + y) a(x + y) &#x2013; 4(x + y) = (x + y)(a &#x2013; 4) b. Factor (2x &#x2013; 3)3x &#x2013; 2(2x &#x2013; 3)
• 49. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) &#x2013; 4(x + y) Pull out the common factor (x + y) a(x + y) &#x2013; 4(x + y) = (x + y)(a &#x2013; 4) b. Factor (2x &#x2013; 3)3x &#x2013; 2(2x &#x2013; 3) Pull out the common factor (2x &#x2013; 3),
• 50. Factoring Out GCF Example D. Factor out the GCF. a. xy &#x2013; 4y = y(x &#x2013; 4) (the GCF is y) b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) (the GCF is 2a) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) (the GCF is 6x2y2) We may pull out common factors that are ( )'s. Example E. Factor a. a(x + y) &#x2013; 4(x + y) Pull out the common factor (x + y) a(x + y) &#x2013; 4(x + y) = (x + y)(a &#x2013; 4) b. Factor (2x &#x2013; 3)3x &#x2013; 2(2x &#x2013; 3) Pull out the common factor (2x &#x2013; 3), (2x &#x2013; 3)3x &#x2013; 2(2x &#x2013; 3) = (2x &#x2013; 3)(3x &#x2013; 2)
• 51. Factoring Out GCF We sometimes pull out a negative sign from an expression.
• 52. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5.
• 53. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5)
• 54. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice.
• 55. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5
• 56. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5 = y(2x &#x2013; 5) &#x2013; (2x &#x2013; 5)
• 57. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5 = y(2x &#x2013; 5) &#x2013; (2x &#x2013; 5) = (2x &#x2013; 5)(y &#x2013; 1)
• 58. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5 = y(2x &#x2013; 5) &#x2013; (2x &#x2013; 5) = (2x &#x2013; 5)(y &#x2013; 1) b. 3x &#x2013; 3y + ax &#x2013; ay
• 59. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5 = y(2x &#x2013; 5) &#x2013; (2x &#x2013; 5) = (2x &#x2013; 5)(y &#x2013; 1) b. 3x &#x2013; 3y + ax &#x2013; ay Group them into two groups. = (3x &#x2013; 3y) + (ax &#x2013; ay)
• 60. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5 = y(2x &#x2013; 5) &#x2013; (2x &#x2013; 5) = (2x &#x2013; 5)(y &#x2013; 1) b. 3x &#x2013; 3y + ax &#x2013; ay Group them into two groups. = (3x &#x2013; 3y) + (ax &#x2013; ay) Factor out the GCF of each group. = 3(x &#x2013; y) + a(x &#x2013; y)
• 61. Factoring Out GCF We sometimes pull out a negative sign from an expression. Example F. Pull out the negative sign from &#x2013;2x + 5. &#x2013;2x + 5 = &#x2013;(2x &#x2013; 5) Sometime we have to use factor twice. Example G. Factor by pulling out twice. a. y(2x &#x2013; 5) &#x2013; 2x + 5 = y(2x &#x2013; 5) &#x2013; (2x &#x2013; 5) = (2x &#x2013; 5)(y &#x2013; 1) b. 3x &#x2013; 3y + ax &#x2013; ay Group them into two groups. = (3x &#x2013; 3y) + (ax &#x2013; ay) Factor out the GCF of each group. = 3(x &#x2013; y) + a(x &#x2013; y) Pull the factor (x &#x2013; y) again. = (3 + a)(x &#x2013; y)