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# 5 1factoring out gcf

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### 5 1factoring out gcf

1. 1. Factoring Out GCF
2. 2. There are three boxes A, B, and C as shown here. Factoring Out GCF
3. 3. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same.
4. 4. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same. For example, we may take two apples from each box,
5. 5. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same. For example, we may take two apples from each box, or three bananas from each box,
6. 6. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same. For example, we may take two apples from each box, or three bananas from each box, or two bananas and two carrot.
7. 7. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same. For example, we may take two apples from each box, or three bananas from each box, or two bananas and two carrot. A group of items which may be taken from each of the three boxes is a group of common items.
8. 8. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same. For example, we may take two apples from each box, or three bananas from each box, or two bananas and two carrot. A group of items which may be taken from each of the three boxes is a group of common items. In this case the largest group of items which may be taken from each of the three boxes consists of
9. 9. There are three boxes A, B, and C as shown here. Factoring Out GCF We are to take items from all three boxes and the items taken from the boxes must be the same. For example, we may take two apples from each box, or three bananas from each box, or two bananas and two carrot. A group of items which may be taken from each of the three boxes is a group of common items. In this case the largest group of items which may be taken from each of the three boxes consists of We define the “greatest common factor” in a similar way.
10. 10. To factor means to rewrite a quantity as a product (without using 1). Factoring Out GCF
11. 11. Example A. Factor 12 completely. To factor means to rewrite a quantity as a product (without using 1). Factoring Out GCF
12. 12. Example A. Factor 12 completely. 12 = 3 * 4 To factor means to rewrite a quantity as a product (without using 1). Factoring Out GCF
13. 13. Example A. Factor 12 completely. 12 = 3 * 4 To factor means to rewrite a quantity as a product (without using 1). A quantity x that can’t be written as product besides as 1*x is said to be prime. Factoring Out GCF
14. 14. Example A. Factor 12 completely. 12 = 3 * 4 To factor means to rewrite a quantity as a product (without using 1). A quantity x that can’t be written as product besides as 1*x is said to be prime. Factoring Out GCF The numbers 2, 3, 5, 7,. . 11, . . are prime.
15. 15. Example A. Factor 12 completely. 12 = 3 * 4 To factor means to rewrite a quantity as a product (without using 1). 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 The numbers 2, 3, 5, 7,. . 11, . . are prime.
16. 16. Example A. Factor 12 completely. 12 = 3 * 4 To factor means to rewrite a quantity as a product (without using 1). 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 not prime The numbers 2, 3, 5, 7,. . 11, . . are prime.
17. 17. Example A. Factor 12 completely. 12 = 3 * 4 = 3 * 2 * 2 To factor means to rewrite a quantity as a product (without using 1). 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 factored completelynot prime The numbers 2, 3, 5, 7,. . 11, . . are prime.
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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
21. 21. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
22. 22. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
23. 23. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
24. 24. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
25. 25. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
26. 26. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
27. 27. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
28. 28. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
29. 29. 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 (without using 1). 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
30. 30. 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 (without using 1). 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, .. The common factor may be a formula in in parenthesis: A common factor of two or more quantities is a factor belongs to all the quantities. The numbers 2, 3, 5, 7,. . 11, . . are prime.
31. 31. 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 (without using 1). 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, .. The common factor may be a formula in in parenthesis: 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. The numbers 2, 3, 5, 7,. . 11, . . are prime.
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 Similar to selecting fruits from boxes, it’s the largest group of common items (factors).
33. 33. 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}
34. 34. 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} The GCF of a list of numbers is the largest number that may be divided by all the number.
35. 35. 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. The GCF of a list of numbers is the largest number that may be divided by all the number.
36. 36. 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} The GCF of a list of numbers is the largest number that may be divided by all the number.
37. 37. 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. The GCF of a list of numbers is the largest number that may be divided by all the number.
38. 38. 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} The GCF of a list of numbers is the largest number that may be divided by all the number.
39. 39. 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. The GCF of a list of numbers is the largest number that may be divided by all the number.
40. 40. 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} = The GCF of a list of numbers is the largest number that may be divided by all the number.
41. 41. 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 GCF of a list of numbers is the largest number that may be divided by all the number.
42. 42. 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. The GCF of a list of numbers is the largest number that may be divided by all the number.
43. 43. 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) The GCF of a list of numbers is the largest number that may be divided by all the number.
44. 44. 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, then factor the “left over” if it’s needed. The GCF of a list of numbers is the largest number that may be divided by all the number.
45. 45. Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y
46. 46. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y
47. 47. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) or (x - 4)y
48. 48. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) or (x - 4)y b. 4ab + 6a
49. 49. (the GCF is y) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3)
50. 50. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3)
51. 51. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
52. 52. (the GCF is y) (the GCF is 2a) Factoring Out GCF Example D. Factor out the GCF. a. xy – 4y = y(x – 4) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2
53. 53. (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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1)
54. 54. (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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
55. 55. 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
56. 56. 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) = (a – 4)(x + y) 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
57. 57. 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) = (a – 4)(x + y) 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
58. 58. 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) = (a – 4)(x + y) 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
59. 59. 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) = (a – 4)(x + y) 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
60. 60. 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) = (a – 4)(x + y) 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
61. 61. 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) = (a – 4)(x + y) 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) or (x - 4)y b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3) c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Note the order of the “( )’s” doesn’t matter because AB=BA.
62. 62. Factor by Grouping There are special four–term formulas where we have to separate the terms into two pairs,
63. 63. There are special four–term formulas where we have to separate the terms into two pairs, Example F. Factor by pulling out twice. a. 3x – 3y + ax – ay Factor by Grouping
64. 64. There are special four–term formulas where we have to separate the terms into two pairs, Example F. Factor by pulling out twice. a. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor by Grouping
65. 65. There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor by Grouping
66. 66. There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 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) Factor by Grouping
67. 67. There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 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) Factor by Grouping A common parenthesis–factor
68. 68. There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 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) Factor by Grouping
69. 69. We may need to pull out the negative sign e.g. writing –4x + 10 as –(2x – 5), in the expression to reveal the common factor. b. y(2x – 5) – 4x + 10 There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 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) Factor by Grouping
70. 70. We may need to pull out the negative sign e.g. writing –4x + 10 as –(2x – 5), in the expression to reveal the common factor. b. y(2x – 5) – 4x + 10 = y(2x – 5) – 2(2x – 5) There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 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) Factor by Grouping
71. 71. We may need to pull out the negative sign e.g. writing –4x + 10 as –(2x – 5), in the expression to reveal the common factor. b. y(2x – 5) – 4x + 10 = y(2x – 5) – 2(2x – 5) = (y – 2) (2x – 5) There are special four–term formulas where we have to separate the terms into two pairs, factor out each pair’s GCF to reveal a common parenthesis–factor, then we pull out this common parenthesis. Example F. Factor by pulling out twice. a. 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) Factor by Grouping
72. 72. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. Factor by Grouping
73. 73. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c Factor by Grouping
74. 74. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. Factor by Grouping
75. 75. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Factor by Grouping
76. 76. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Example G. Factor the the trinomial x2 – 3x + 2 by grouping given that x2 – 3x + 2 = x2 – 2x – x + 2 Factor by Grouping
77. 77. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Example G. Factor the the trinomial x2 – 3x + 2 by grouping given that x2 – 3x + 2 = x2 – 2x – x + 2 Group x2 – 2x – x + 2 into two groups. x2 – 2x – x + 2 Factor by Grouping
78. 78. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Example G. Factor the the trinomial x2 – 3x + 2 by grouping given that x2 – 3x + 2 = x2 – 2x – x + 2 Group x2 – 2x – x + 2 into two groups. x2 – 2x – x + 2 = (x2 – 2x) + (–x + 2) Factor by Grouping
79. 79. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Example G. Factor the the trinomial x2 – 3x + 2 by grouping given that x2 – 3x + 2 = x2 – 2x – x + 2 Group x2 – 2x – x + 2 into two groups. x2 – 2x – x + 2 = (x2 – 2x) + (–x + 2) Factor out the GCF of each group. = x(x – 2) –1(x – 2) Factor by Grouping
80. 80. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Example G. Factor the the trinomial x2 – 3x + 2 by grouping given that x2 – 3x + 2 = x2 – 2x – x + 2 Group x2 – 2x – x + 2 into two groups. x2 – 2x – x + 2 = (x2 – 2x) + (–x + 2) Factor out the GCF of each group. = x(x – 2) –1(x – 2) Pull the factor (x – 2) again. = (x – 2 )(x – 1) Factor by Grouping
81. 81. Trinomials (three-term) are polynomials of the form ax2 + bx + c where a, b, and c are numbers. The product of two binomials is a trinomials: (#x + #)(#x + #)  ax2 + bx + c For example (x + 2)(2x + 3) = 2x2 + 7x + 6. To factor a trinomial means to a undo the multiplication and write the trinomial as a product of two binomials, if possible. ax2 + bx + c  (#x + #)(#x + #) Example G. Factor the the trinomial x2 – 3x + 2 by grouping given that x2 – 3x + 2 = x2 – 2x – x + 2 Group x2 – 2x – x + 2 into two groups. x2 – 2x – x + 2 = (x2 – 2x) + (–x + 2) Factor out the GCF of each group. = x(x – 2) –1(x – 2) Pull the factor (x – 2) again. = (x – 2 )(x – 1) We will use grouping as the default method for factoring trinomials. Factor by Grouping
82. 82. 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
83. 83. 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
84. 84. 35. –3x3 – 30x2 – 48x34. –yx2 + 4yx + 5y 36. –2x3 + 20x2 – 24x 40. 4x2 – 44xy + 96y2 37. –x2 + 11xy + 24y2 38. x4 – 6x3 + 36x2 39. –x2 + 9xy + 36y2 C. Factor. Factor out the GCF, the “–”, and arrange the terms in order first. D. Factor. If not possible, state so. 41. x2 + 1 42. x2 + 4 43. x2 + 9 43. 4x2 + 25 44. What can you conclude from 41–43? Factoring Trinomials and Making Lists
85. 85. Exercise. A. Factor. If it’s prime, state so. 1. x2 – x – 2 2. x2 + x – 2 3. x2 – x – 6 4. x2 + x – 6 5. x2 – x + 2 6. x2 + 2x – 3 7. x2 + 2x – 8 8. x2 – 3x – 4 9. x2 + 5x + 6 10. x2 + 5x – 6 13. x2 – x – 20 11. x2 – 5x – 6 12. x2 – 5x + 6 17. x2 – 10x – 24 14. x2 – 8x – 20 15. x2 – 9x – 20 16. x2 – 9x + 20 18. x2 – 10x + 24 19. x2 – 11x + 24 20. x2 – 11x – 24 21. x2 – 12x – 36 22. x2 – 12x + 36 23. x2 – 13x – 36 24. x2 – 13x + 36 B. Factor. Factor out the GCF, the “–”, and arrange the terms in order first if necessary. 29. 3x2 – 30x – 7227. –x2 – 5x + 14 28. 2x3 – 18x2 + 40x 30. –2x3 + 20x2 – 24x 25. x2 – 36 26. x2 + 36 31. –2x4 + 18x2 32. –3x – 24x3 + 22x2 33. 5x4 + 10x5 – 5x3 Factoring Trinomials and Making Lists