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### 1 1 review on factoring

1. 1. Review on Factoring Frank Ma © 2011
2. 2. To factor means to rewrite an expression as a product in a nontrivial way. Review on Factoring
3. 3. To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring
4. 4. To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring I. Always pull out the greatest common factor first.
5. 5. To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method.
6. 6. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
7. 7. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
8. 8. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
9. 9. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
10. 10. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 3 is a common factor. A common factor of two or more quantities is a factor that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
11. 11. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 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 that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
12. 12. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 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 that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
13. 13. ± ± – + To factor means to rewrite an expression as a product in a nontrivial way. Following are the steps to factor an expression. Review on Factoring Example A. a. Since 6 = (2)(3) and 15 = (3)(5), so 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 that belongs to all the quantities. Pulling out GCF I. Always pull out the greatest common factor first. II. If the expression is a trinomial (three-term) ax2 + bx + c, use the reverse-FOIL method or the ac-method. III. Use the following factoring formulas if possible x2 – y2 = (x + y)(x – y). x3 y3 = ( x y )( x2 xy + y2 ).
14. 14. The greatest common factor (GCF) is the common factor that has the largest coefficient and highest degree of each factor among all common factors. Review on Factoring
15. 15. 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 B. Find the GCF of the given quantities. a. GCF{24, 36} Review on Factoring
16. 16. 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 B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. Review on Factoring
17. 17. 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 B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} Review on Factoring
18. 18. 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 B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. Review on Factoring
19. 19. 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 B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} Review on Factoring
20. 20. 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 B. Find the GCF of the given quantities. a. GCF{24, 36} = 12. b. GCF{4ab, 6a} = 2a. c. GCF {6xy2, 15 x2y2} = 3xy2. Review on Factoring
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. Example B. 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} = Review on Factoring
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. Example B. 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. Review on Factoring
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. Example B. 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 backwards gives the Extraction Law, that is, common factors may be extracted from sums or differences. Review on Factoring
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. Example B. 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 backwards gives the Extraction Law, that is, common factors may be extracted from sums or differences. AB ± AC  A(B±C) Review on Factoring
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. Example B. 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 backwards 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 common factor”. To factor, the first step always is to factor out the GCF. Review on Factoring
26. 26. Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 Review on Factoring
27. 27. (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 Review on Factoring
28. 28. (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) Review on Factoring
29. 29. (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
30. 30. b. (2x – 3)3x – 2(2x – 3) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
31. 31. b. (2x – 3)3x – 2(2x – 3) Pull out the common factor (2x – 3), (2x – 3)3x – 2(2x – 3) (the GCF is 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
32. 32. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) Review on Factoring
33. 33. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Review on Factoring
34. 34. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Review on Factoring
35. 35. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring
36. 36. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring
37. 37. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring d. x2 – x – 6
38. 38. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6
39. 39. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6)
40. 40. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factor of each = x(x – 3) + 2(x – 3)
41. 41. b. (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 6x2y2) Example C. Factor out the GCF. a. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1) c. 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) Review on Factoring d. x2 – x – 6 Write – x as –3x + 2x = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factor of each = x(x – 3) + 2(x – 3) Take out the common (x – 3) = (x – 3)(x + 2)
42. 42. Factoring Trinomials and Making Lists
43. 43. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. Factoring Trinomials and Making Lists
44. 44. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. Factoring Trinomials and Making Lists
45. 45. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Factoring Trinomials and Making Lists
46. 46. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists
47. 47. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials:
48. 48. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1)
49. 49. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1) ll. the ones that are prime or no factorable, such as x2 + 2x + 3  (#x + #)(#x + #) (Not possible!)
50. 50. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1) ll. the ones that are prime or no factorable, such as x2 + 2x + 3  (#x + #)(#x + #) Our jobs are to determine which trinomials: 1. are factorable and factor them, (Not possible!)
51. 51. For our discussions, trinomials (three-term) in x are polynomials of the form ax2 + bx + c where a (≠ 0), b, and c are numbers. In general, we have that (#x + #)(#x + #)  ax2 + bx + c. For example, (x + 2)(x + 1)  x2 + 3x + 2 with a = 1, b = 3, and c = 2. Hence, "to factor a trinomial" means to write the trinomial as a product of two binomials, that is, to convert ax2 + bx + c  (#x + #)(#x + #) Factoring Trinomials and Making Lists The Basic Fact About Factoring Trinomials: There are two types of trinomials, l. the ones that are factorable such as x2 + 3x + 2  (x + 2)(x + 1) ll. the ones that are prime or no factorable, such as x2 + 2x + 3  (#x + #)(#x + #) Our jobs are to determine which trinomials: 1. are factorable and factor them, 2. are prime so we won’t waste time on trying to factor them. (Not possible!)
52. 52. Factoring Trinomials and Making Lists One method to determine which is which is by making lists.
53. 53. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”.
54. 54. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers.
55. 55. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 12 I II Example D. Using the given tables, list all the u and v such that: 7 9
56. 56. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 12 I i. uv is the top number II Example D. Using the given tables, list all the u and v such that: 7 9
57. 57. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 12 9 I ii. and if possible, u + v is the bottom number. i. uv is the top number II Example D. Using the given tables, list all the u and v such that:
58. 58. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II Example D. Using the given tables, list all the u and v such that:
59. 59. Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. For l, the solution are 3 and 4. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II Example D. Using the given tables, list all the u and v such that:
60. 60. Example D. Using the given tables, list all the u and v such that: Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. For l, the solution are 3 and 4. For ll, based on the list of all the possible u and v, there are no u and v where u + v = 9, so the task is impossible. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II
61. 61. Example D. Using the given tables, list all the u and v such that: Factoring Trinomials and Making Lists One method to determine which is which is by making lists. A list is a record of all the possibilities according to some criteria such as the list of “all the cousins that one has”. The lists we will make are lists of numbers. 12 7 We list all the possible ways to factor 12 as u*v as shown. For l, the solution are 3 and 4. For ll, based on the list of all the possible u and v, there are no u and v where u + v = 9, so the task is impossible. 12 9 I 1 12 6 3 4 2 1 12 6 3 4 2 ii. and if possible, u + v is the bottom number. i. uv is the top number II impossible!
62. 62. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Factoring Trinomials and Making Lists
63. 63. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
64. 64. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
65. 65. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
66. 66. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
67. 67. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) take out the common (x – 3) = (x – 3)(x + 2) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
68. 68. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. II. If the table is impossible to do, then the trinomial is prime. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) take out the common (x – 3) = (x – 3)(x + 2) Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
69. 69. The ac-Method A table like the ones above can be made from a given trinomial and the ac–method uses the table to check if the given trinomial is factorable or prime. I. If we find the u and v that fit the table then it is factorable, and we may use the grouping method, with the found u and v, to factor the trinomial. II. If the table is impossible to do, then the trinomial is prime. Example G. Factor x2 – x – 6 by grouping. x2 – x – 6 write –x as –3x + 2x = x2 – 3x + 2x – 6 put the four terms into two pairs = (x2 – 3x) + (2x – 6) take out the GCF of each pair = x(x – 3) + 2(x – 3) take out the common (x – 3) = (x – 3)(x + 2) Here is an example of factoring a trinomial by grouping. Here is how the X–table is made from a trinomial. Factoring Trinomials and Making Lists
70. 70. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: ac b Factoring Trinomials and Making Lists
71. 71. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, ac b Factoring Trinomials and Making Lists
72. 72. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, In example B, the ac-table for 1x2 – x – 6 is: ac b Factoring Trinomials and Making Lists –6 –1
73. 73. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that uv = ac u + v = b In example B, the ac-table for 1x2 – x – 6 is: ac b Factoring Trinomials and Making Lists –6 –1 u v
74. 74. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, ac b Factoring Trinomials and Making Lists u v
75. 75. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, ac b Factoring Trinomials and Making Lists u v
76. 76. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 ac b Factoring Trinomials and Making Lists u v
77. 77. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 ac b Factoring Trinomials and Making Lists u v
78. 78. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping = (x2 – 3x) + (2x – 6) ac b Factoring Trinomials and Making Lists u v
79. 79. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping = (x2 – 3x) + (2x – 6) = x(x – 3) + 2(x – 3) ac b u v Factoring Trinomials and Making Lists
80. 80. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: i.e. ac at the top, and b at the bottom, and we are to find u and v such that –6 –1 –3 2 uv = ac u + v = b I. If u and v are found (so u + v = b), write ax2 + bx + c as ax2 + ux + vx + c, then factor (ax2 + ux) + (vx + c) by the grouping method. In example B, the ac-table for 1x2 – x – 6 is: We found –3, 2 fit the table, so we write x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping = (x2 – 3x) + (2x – 6) = x(x – 3) + 2(x – 3) = (x – 3)(x + 2) ac b u v Factoring Trinomials and Making Lists
81. 81. Example H. Factor 3x2 – 4x – 20 using the ac-method. Factoring Trinomials and Making Lists
82. 82. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, Factoring Trinomials and Making Lists
83. 83. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: –60 –4 Factoring Trinomials and Making Lists
84. 84. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. –60 –4 Factoring Trinomials and Making Lists
85. 85. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists
86. 86. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
87. 87. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups
88. 88. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor
89. 89. Example H. Factor 3x2 – 4x – 20 using the ac-method. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) pull out common factor Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)
90. 90. Example H. Factor 3x2 – 4x – 20 using the ac-method. If the trinomial is prime then we have to justify it’s prime by showing that no such u and v exist We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) pull out common factor Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)
91. 91. Example H. Factor 3x2 – 4x – 20 using the ac-method. If the trinomial is prime then we have to justify it’s prime by showing that no such u and v exist by listing all the possible u’s and v’s such that uv = ac in the table to demonstrate that none of them fits the condition u + v = b. We have that a = 3, c = –20 so ac = 3(–20) = –60, b = –4 and the ac–table is: We need two numbers u and v such that uv = –60 and u + v = –4. By trial and error we see that 6 and –10 is the solution so we may factor the trinomial by grouping. –60 –4 –10 6 Factoring Trinomials and Making Lists Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20 = (3x2 + 6x ) + (–10x – 20) put in two groups = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) pull out common factor Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)
92. 92. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. Factoring Trinomials and Making Lists
93. 93. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: Factoring Trinomials and Making Lists –60 –6
94. 94. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. Factoring Trinomials and Making Lists –60 –6
95. 95. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. Factoring Trinomials and Making Lists –60 –6
96. 96. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. Factoring Trinomials and Making Lists –60 –6 601 302 203 154 125 106 Always make a list in an orderly manner to ensure the accuracy of the list.
97. 97. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. By the table, we see that there are no u and v such that ±u and ±v combine to be –6. Hence 3x2 – 6x – 20 is prime. Factoring Trinomials and Making Lists –60 –6 601 302 203 154 125 106 Always make a list in an orderly manner to ensure the accuracy of the list.
98. 98. Example I. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. a = 3, c = –20, hence ac = 3(–20) = –60, with b = –6, we have the ac–table: We want two numbers u and v such that uv = –60 and u + v = –6. After failing to guess two such numbers, we check to see if it's prime by listing in order all positive u’s and v’s where uv = 60 as shown. By the table, we see that there are no u and v such that ±u and ±v combine to be –6. Hence 3x2 – 6x – 20 is prime. Factoring Trinomials and Making Lists –60 –6 601 302 203 154 125 106 Always make a list in an orderly manner to ensure the accuracy of the list. Finally for some trinomials, such as when a = 1 or x2 + bx + c, it’s easier to guess directly because it must factor into the form (x ± u) (x ± v) if it’s factorable.
99. 99. Example J. a. Factor x2 + 5x + 6 Factoring Trinomials and Making Lists
100. 100. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 Factoring Trinomials and Making Lists
101. 101. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Factoring Trinomials and Making Lists
102. 102. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) Factoring Trinomials and Making Lists
103. 103. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x Factoring Trinomials and Making Lists
104. 104. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, Factoring Trinomials and Making Lists
105. 105. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
106. 106. b. Factor x2 – 5x + 6 Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
107. 107. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
108. 108. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
109. 109. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
110. 110. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
111. 111. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
112. 112. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
113. 113. c. Factor x2 + 5x – 6 b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
114. 114. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
115. 115. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
116. 116. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
117. 117. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) and –1 + 6 = 5, b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
118. 118. c. Factor x2 + 5x – 6 We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and u + v = 5. Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) and –1 + 6 = 5, so x2 + 5x – 6 = (x – 1)(x + 6). b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 and u + v = –5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5, so x2 – 5x + 6 = (x – 2)(x – 3). Example J. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 and u + v = 5. Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x, so x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
119. 119. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y).
120. 120. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2
121. 121. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2
122. 122. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y)
123. 123. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x
124. 124. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1)
125. 125. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1)
126. 126. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1)
127. 127. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) =
128. 128. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1)
129. 129. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1) = 902 – 12
130. 130. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1) = 902 – 12 = 8,100 – 1 = 7,099
131. 131. Review on Factoring Factoring Formula If it fits, use the Difference of Squares Formula x2 – y2 = (x + y)(x – y). Example K. a. 4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y)(2x + 3y) b. 32x3 – 2x = 2x(16x2 – 1) = 2x(4x + 1)(4x – 1) c. (89)(91) = (90 – 1)(90 + 1) = 902 – 12 = 8,100 – 1 = 7,099 The factors (x + y) and (x – y) are called the conjugate of each other.
132. 132. Ex. A. Factor the following trinomials. use any method. If it’s prime, state so. 1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x – 1 4. 3x2 + 2x – 1 5. 2x2 – 3x + 1 6. 2x2 + 3x – 1 8. 2x2 – 3x – 27. 2x2 + 3x – 2 15. 6x2 + 5x – 6 10. 5x2 + 9x – 2 Ex. B. Factor. Factor out the GCF, the “–”, and arrange the terms in order first. 9. 5x2 – 3x – 2 12. 3x2 – 5x + 211. 3x2 + 5x + 2 14. 6x2 – 5x – 613. 3x2 – 5x + 2 16. 6x2 – x – 2 17. 6x2 – 13x + 2 18. 6x2 – 13x + 2 19. 6x2 + 7x + 2 20. 6x2 – 7x + 2 21. 6x2 – 13x + 6 22. 6x2 + 13x + 6 23. 6x2 – 5x – 4 24. 6x2 – 13x + 8 25. 6x2 – 13x – 8 26. 4x2 – 49 27. 25x2 – 4 28. 4x2 + 9 29. 25x2 + 9 30. – 6x2 – 5xy + 6y2 31. – 3x2 + 2x3– 2x 32. –6x3 – x2 + 2x 33. –15x3 – 25x2 – 10x 34. 12x3y2 –14x2y2 + 4xy2 Review on Factoring