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# 5 2factoring trinomial i

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### 5 2factoring trinomial i

1. 1. Factoring Trinomials and Making Lists
2. 2. 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
3. 3. 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 (#x + #)(#x + #)  ax2 + bx + c. Factoring Trinomials and Making Lists
4. 4. 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 (#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
5. 5. 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 (#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
6. 6. 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 (#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:
7. 7. 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 (#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)
8. 8. 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 (#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!)
9. 9. 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 (#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!)
10. 10. 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 (#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!)
11. 11. Factoring Trinomials and Making Lists One method to determine which is which is by making lists.
12. 12. 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”.
13. 13. 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.
14. 14. Example A. 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 12 I i. uv is the top number II
15. 15. Example A. 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 12 9 I ii. and if possible, u + v is the bottom number. i. uv is the top number II
16. 16. Example A. 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. 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
17. 17. Example A. 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. 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
18. 18. Example A. 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
19. 19. 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
20. 20. 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 B. Factor x2 – x – 6 by grouping. Here is an example of factoring a trinomial by grouping. Factoring Trinomials and Making Lists
21. 21. 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 B. 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
22. 22. 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 B. 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
23. 23. 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 B. 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
24. 24. 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 B. 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
25. 25. 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 B. 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
26. 26. 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 B. 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. Let’s see how the X–table is made from a trinomial. Factoring Trinomials and Making Lists
27. 27. ac-Method: Given the trinomial ax2 + bx + c with no common factor, it’s ac–table is: ac b Factoring Trinomials and Making Lists
28. 28. 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
29. 29. 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
30. 30. 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
31. 31. 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
32. 32. 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
33. 33. 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
34. 34. 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
35. 35. 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
36. 36. 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
37. 37. Example C. Factor 3x2 – 4x – 20 using the ac-method. Factoring Trinomials and Making Lists
38. 38. Example C. 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
39. 39. Example C. 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
40. 40. Example C. 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
41. 41. Example C. 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
42. 42. Example C. 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
43. 43. Example C. 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
44. 44. Example C. 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
45. 45. Example C. 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)
46. 46. Example C. 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)
47. 47. Example C. 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)
48. 48. Example D. Factor 3x2 – 6x – 20 if possible. If it’s prime, justify that. Factoring Trinomials and Making Lists
49. 49. Example D. 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
50. 50. Example D. 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
51. 51. Example D. 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
52. 52. Example D. 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.
53. 53. Example D. 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. 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.
54. 54. Example D. 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.
55. 55. Example D. 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. Factoring By Trial and Error
56. 56. Example D. 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. Factoring By Trial and Error
57. 57. Example E. a. Factor x2 + 5x + 6 By Trial and Error
58. 58. Example E. a. Factor x2 + 5x + 6 We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v where uv = 6 By Trial and Error
59. 59. Example E. 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. By Trial and Error
60. 60. Example E. 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) By Trial and Error
61. 61. Example E. 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 By Trial and Error
62. 62. Example E. 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, By Trial and Error
63. 63. Example E. 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) By Trial and Error
64. 64. b. Factor x2 – 5x + 6 Example E. 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) By Trial and Error
65. 65. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, Example E. 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) By Trial and Error
66. 66. b. Factor x2 – 5x + 6 We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v where uv = 6 Example E. 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) By Trial and Error
67. 67. 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 E. 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) By Trial and Error
68. 68. 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 E. 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) By Trial and Error
69. 69. 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 E. 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) By Trial and Error
70. 70. 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 E. 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) By Trial and Error
71. 71. 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 E. 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) By Trial and Error
72. 72. 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 E. 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) By Trial and Error
73. 73. 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 E. 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) By Trial and Error
74. 74. 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 E. 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) By Trial and Error
75. 75. 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 E. 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) By Trial and Error
76. 76. 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 E. 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) By Trial and Error
77. 77. This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options. By Trial and Error
78. 78. Example F. Factor 3x2 + 5x + 2. By Trial and Error This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
79. 79. Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). By Trial and Error This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
80. 80. Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 By Trial and Error This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
81. 81. Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. By Trial and Error This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
82. 82. Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. By Trial and Error This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
83. 83. Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, By Trial and Error This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
84. 84. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. By Trial and Error Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
85. 85. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, By Trial and Error Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
86. 86. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, By Trial and Error Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
87. 87. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, By Trial and Error Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
88. 88. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, we see that 3x2 + 5x + 2 = (3x + 2)(1x + 1). By Trial and Error Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
89. 89. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, we see that 3x2 + 5x + 2 = (3x + 2)(1x + 1). 5x By Trial and Error Example F. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that This reversed–FOIL method (by trial and error) is useful when the numbers involved can only be factored in few options.
90. 90. Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists
91. 91. Example G. Use b2 – 4ac to check if the trinomial is factorable. a. 3x2 – 7x + 2 Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2
92. 92. Example G. Use b2 – 4ac to check if the trinomial is factorable. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. a. 3x2 – 7x + 2 Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2
93. 93. Example G. Use b2 – 4ac to check if the trinomial is factorable. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. a. 3x2 – 7x + 2 Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2 a = 3, b = (–7) and c = 2
94. 94. Example G. Use b2 – 4ac to check if the trinomial is factorable. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. a. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists a = 3, b = (–7) and c = 2 b. 3x2 – 7x – 2
95. 95. Example G. Use b2 – 4ac to check if the trinomial is factorable. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. a. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2 a = 3, b = (–7) and c = 2
96. 96. Example G. Use b2 – 4ac to check if the trinomial is factorable. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. If b2 – 4ac = not a squared number, then it’s not factorable. a. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2 a = 3, b = (–7) and c = 2
97. 97. Example G. Use b2 – 4ac to check if the trinomial is factorable. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. If b2 – 4ac = not a squared number, then it’s not factorable. a. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2 a = 3, b = (–7) and c = 2 a = 3, b = (–7) and c = (–2)
98. 98. Example G. Use b2 – 4ac to check if the trinomial is factorable. b2 – 4ac = (–7)2 – 4(3)(–2) Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. If b2 – 4ac = not a squared number, then it’s not factorable. a. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2 a = 3, b = (–7) and c = 2 a = 3, b = (–7) and c = (–2)
99. 99. Example G. Use b2 – 4ac to check if the trinomial is factorable. b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. Theorem: The trinomial ax2 + bx + c is factorable if b2 – 4ac = 0, 1, 4, 9, 16, 25, ..i.e. it’s a squared number. If b2 – 4ac = not a squared number, then it’s not factorable. a. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. Besides the ac–method, here is another method that’s based on a calculating a number to check if a trinomial is factorable. Factoring Trinomials and Making Lists b. 3x2 – 7x – 2 a = 3, b = (–7) and c = 2 a = 3, b = (–7) and c = (–2)
100. 100. Observations About Signs Factoring Trinomials and Making Lists
101. 101. Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. Factoring Trinomials and Making Lists
102. 102. Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. Factoring Trinomials and Making Lists
103. 103. Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. From the examples above x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
104. 104. Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. if b is negative, then both are negative. From the examples above x2 + 5x + 6 = (x + 2)(x + 3) Factoring Trinomials and Making Lists
105. 105. { Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. if b is negative, then both are negative. From the examples above x2 + 5x + 6 = (x + 2)(x + 3) x2 – 5x + 6 = (x – 2)(x – 3) Factoring Trinomials and Making Lists
106. 106. { Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. if b is negative, then both are negative. From the examples above x2 + 5x + 6 = (x + 2)(x + 3) x2 – 5x + 6 = (x – 2)(x – 3) 2. If c is negative, then u and v have opposite signs. Factoring Trinomials and Making Lists
107. 107. { Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. if b is negative, then both are negative. From the examples above x2 + 5x + 6 = (x + 2)(x + 3) x2 – 5x + 6 = (x – 2)(x – 3) 2. If c is negative, then u and v have opposite signs. The one with larger absolute value has the same sign as b. Factoring Trinomials and Making Lists
108. 108. { Observations About Signs Given that x2 + bx + c = (x + u)(x + v) so that uv = c, we observe the following. 1. If c is positive, then u and v have same sign. In particular, if b is also positive, then both are positive. if b is negative, then both are negative. From the examples above x2 + 5x + 6 = (x + 2)(x + 3) x2 – 5x + 6 = (x – 2)(x – 3) 2. If c is negative, then u and v have opposite signs. The one with larger absolute value has the same sign as b. From the example above x2 – 5x – 6 = (x – 6)(x + 1) Factoring Trinomials and Making Lists
109. 109. 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 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 25. 4x2 – 9 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 Exercise A. Use the ac–method, factor the trinomial or demonstrate that it’s not factorable. Factoring Trinomials and Making Lists
110. 110. 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