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# 5 4factoring trinomial ii

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### 5 4factoring trinomial ii

1. 1. Factoring Trinomials II<br />
2. 2. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />
3. 3. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. <br />
4. 4. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />
5. 5. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />
6. 6. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />
7. 7. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. <br />
8. 8. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />
9. 9. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />
10. 10. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />
11. 11. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />Yes, 1* (2) + 3 * (1) = 5<br />
12. 12. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />Yes, 1* (2) + 3 * (1) = 5<br />b. 1* (± ) + 3* (± ) = –5.<br />
13. 13. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />Yes, 1* (2) + 3 * (1) = 5<br />b. 1* (± ) + 3* (± ) = –5.<br />Yes, 1* (1) + 3* (–2) = –5<br />
14. 14. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />Yes, 1* (2) + 3 * (1) = 5<br />b. 1* (± ) + 3* (± ) = –5.<br />Yes, 1* (1) + 3* (–2) = –5<br />or 1* (–2) + 3* (–1) = –5<br />
15. 15. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />Yes, 1* (2) + 3 * (1) = 5<br />b. 1* (± ) + 3* (± ) = –5.<br />Yes, 1* (1) + 3* (–2) = –5<br />or 1* (–2) + 3* (–1) = –5<br />c. 1* (± ) + 3* (± ) = 8.<br />
16. 16. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />We’ll give two methods. One is short but not reliable. <br />The second one takes more steps but gives definite answers.<br />Reversed FOIL Method<br />For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. <br />Example A. Let {1, 3} and {1, 2} be two pairs of numbers. <br />Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true?<br />a. 1* (± ) + 3*(± ) = 5.<br />Yes, 1* (2) + 3 * (1) = 5<br />b. 1* (± ) + 3* (± ) = –5.<br />Yes, 1* (1) + 3* (–2) = –5<br />or 1* (–2) + 3* (–1) = –5<br />c. 1* (± ) + 3* (± ) = 8.<br />No, since the most we can obtain is 1* (1) + 3* (2) = 7.<br />
17. 17. Factoring Trinomials II<br />(Reversed FOIL Method) <br />
18. 18. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />
19. 19. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />
20. 20. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />
21. 21. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />
22. 22. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x.<br />
23. 23. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x.<br />That is, (3x ± #)(1x ± #) must yields +5x, <br />
24. 24. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x.<br />That is, (3x ± #)(1x ± #) must yields +5x, or that <br />3(± # ) +1(± #) = 5 where the #’s are 1 and 2. <br />
25. 25. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x.<br />That is, (3x ± #)(1x ± #) must yields +5x, or that <br />3(± # ) +1(± #) = 5 where the #’s are 1 and 2. <br />Since 3(1) +1(2) = 5, <br />
26. 26. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x.<br />That is, (3x ± #)(1x ± #) must yields +5x, or that <br />3(± # ) +1(± #) = 5 where the #’s are 1 and 2. <br />Since 3(1) +1(2) = 5, we see that<br />3x2 + 5x + 2 = (3x + 2)(1x + 1).<br />
27. 27. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />Example B. Factor 3x2 + 5x + 2.<br />The only way to get 3x2 is (3x ± #)(1x ± #). <br />The #’s must be 1 and 2 to get the constant term +2.<br />We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x.<br />That is, (3x ± #)(1x ± #) must yields +5x, or that <br />3(± # ) +1(± #) = 5 where the #’s are 1 and 2. <br />Since 3(1) +1(2) = 5, we see that<br />3x2 + 5x + 2 = (3x + 2)(1x + 1).<br />5x<br />
28. 28. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />
29. 29. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />
30. 30. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />
31. 31. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />
32. 32. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />
33. 33. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />Example D. Factor 3x2 + 5x – 2.<br />
34. 34. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />Example D. Factor 3x2 + 5x – 2.<br />We start with (3x ± #)(1x ± #). <br />
35. 35. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />Example D. Factor 3x2 + 5x – 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 so that<br />3(± # ) + 1(± # ) = +5.<br />
36. 36. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />Example D. Factor 3x2 + 5x – 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 so that<br />3(± # ) + 1(± # ) = +5.<br />Since c is negative, they must have opposite signs .<br />
37. 37. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />Example D. Factor 3x2 + 5x – 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 so that<br />3(± # ) + 1(± # ) = +5.<br />Since c is negative, they must have opposite signs .<br />It is 3(+2) + 1(–1) = +5.<br />
38. 38. Factoring Trinomials II<br />Example C. Factor 3x2 – 7x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 as #'s so that<br />3(± # ) + 1(± # ) = –7.<br />It's 3(–2) + 1(–1) = –7.<br />So 3x2 – 7x + 2 = (3x –1)(1x – 2)<br />Example D. Factor 3x2 + 5x – 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1 and 2 so that<br />3(± # ) + 1(± # ) = +5.<br />Since c is negative, they must have opposite signs .<br />It is 3(+2) + 1(–1) = +5.<br />So 3x2 + 5x + 2 = (3x –1)(1x + 2)<br />
39. 39. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />
40. 40. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #).<br />
41. 41. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />
42. 42. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. <br />
43. 43. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />
44. 44. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />
45. 45. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />
46. 46. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />We start with (3x ± #)(1x ± #). <br />
47. 47. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), <br />we need to fill in 2&2 or 1&4 so that<br />3(± # ) + 1(± # ) = +11. <br />
48. 48. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), <br />we need to fill in 2&2 or 1&4 so that<br />3(± # ) + 1(± # ) = +11. It can't be 2&2. <br />
49. 49. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), <br />we need to fill in 2&2 or 1&4 so that<br />3(± # ) + 1(± # ) = +11. It can't be 2&2. <br />Try 1&4, <br />
50. 50. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), <br />we need to fill in 2&2 or 1&4 so that<br />3(± # ) + 1(± # ) = +11. It can't be 2&2. <br />Try 1&4, it is <br />3(+4) + 1(–1) = +11. <br />
51. 51. Factoring Trinomials II<br />Example E. Factor 3x2 + 8x + 2.<br />We start with (3x ± #)(1x ± #). <br />We need to fill in 1&2 so that<br />3(± # ) + 1(± # ) = +8.<br />This is impossible. Hence the expression is prime.<br />If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.<br />Example F. Factor 3x2 + 11x – 4.<br />We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), <br />we need to fill in 2&2 or 1&4 so that<br />3(± # ) + 1(± # ) = +11. It can't be 2&2. <br />Try 1&4, it is <br />3(+4) + 1(–1) = +11. <br />So 3x2 + 11x – 4 = (3x – 1)(1x + 4).<br />
52. 52. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />
53. 53. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />
54. 54. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), <br />
55. 55. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12<br />
56. 56. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />
57. 57. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />12 = 1(12) = 2(6) = 3(4) <br />
58. 58. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />12 = 1(12) = 2(6) = 3(4) <br />1&12 and 2&6 can be quickly eliminated. <br />
59. 59. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />12 = 1(12) = 2(6) = 3(4) <br />1&12 and 2&6 can be quickly eliminated. <br />We get (3)(–3) + (4)(+1) = – 5.<br />
60. 60. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />12 = 1(12) = 2(6) = 3(4) <br />1&12 and 2&6 can be quickly eliminated. <br />We get (3)(–3) + (4)(+1) = – 5.<br />So 12x2 – 5x – 3 = (3x + 1)(4x – 3).<br />
61. 61. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />12 = 1(12) = 2(6) = 3(4) <br />1&12 and 2&6 can be quickly eliminated. <br />We get (3)(–3) + (4)(+1) = – 5.<br />So 12x2 – 5x – 3 = (3x + 1)(4x – 3).<br />Remark:<br />In the above method, finding<br />(#)(± #) + (#)( ± #) = b <br />does not guarantee that the trinomial will factor. <br />
62. 62. Factoring Trinomials II<br />It's not necessary to always start with ax2. If c is a prime number, we start with c.<br />Example G. Factor 12x2 – 5x – 3.<br />Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that <br />(± #)(± 3) + (± #)(±1) = – 5. <br />12 = 1(12) = 2(6) = 3(4) <br />1&12 and 2&6 can be quickly eliminated. <br />We get (3)(–3) + (4)(+1) = – 5.<br />So 12x2 – 5x – 3 = (3x + 1)(4x – 3).<br />Remark:<br />In the above method, finding<br />(#)(± #) + (#)( ± #) = b <br />does not guarantee that the trinomial will factor. We have to match the sign of c also.<br />
63. 63. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />
64. 64. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #).<br />
65. 65. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />
66. 66. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />
67. 67. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />In fact this trinomial is prime.<br />
68. 68. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />In fact this trinomial is prime.<br />There might be multiple matchings for<br />(#)(± #) + (#)( ± #) = b <br />make sure you chose the correct one, if any.<br />
69. 69. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />In fact this trinomial is prime.<br />There might be multiple matchings for<br />(#)(± #) + (#)( ± #) = b <br />make sure you chose the correct one, if any.<br />Example I: Factor 1x2 + 5x – 6 .<br />
70. 70. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />In fact this trinomial is prime.<br />There might be multiple matchings for<br />(#)(± #) + (#)( ± #) = b <br />make sure you chose the correct one, if any.<br />Example I: Factor 1x2 + 5x – 6 .<br />We have:<br />1(+3) + 1(+2) = +5 <br />
71. 71. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />In fact this trinomial is prime.<br />There might be multiple matchings for<br />(#)(± #) + (#)( ± #) = b <br />make sure you chose the correct one, if any.<br />Example I: Factor 1x2 + 5x – 6 .<br />We have:<br />1(+3) + 1(+2) = +5 <br />1(+6) + 1(–1) = +5 <br />
72. 72. Factoring Trinomials II<br />Example H. Factor 3x2 – 7x – 2 .<br />We start with (3x ± #)(1x ± #). We find that:<br />3(–2) + 1(–1) = –7.<br />But this won't work since (–2)(–1) = 2 = c. <br />In fact this trinomial is prime.<br />There might be multiple matchings for<br />(#)(± #) + (#)( ± #) = b <br />make sure you chose the correct one, if any.<br />Example I: Factor 1x2 + 5x – 6 .<br />We have:<br />1(+3) + 1(+2) = +5 <br />1(+6) + 1(–1) = +5 <br />The one that works is x2 + 5x – 6 = (x + 6)(x – 1).<br />
73. 73. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />
74. 74. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order.<br />
75. 75. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />
76. 76. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />3. make sure that x2 is positive, if not, factor out the negative sign first. <br />
77. 77. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />3. make sure that x2 is positive, if not, factor out the negative sign first. <br />Example J. Factor –x3 + 3x + 2x2<br />
78. 78. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />3. make sure that x2 is positive, if not, factor out the negative sign first. <br />Example J. Factor –x3 + 3x + 2x2<br />–x3 + 3x + 2x2 Arrange the terms in order<br />= –x3 + 2x2 + 3x <br />
79. 79. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />3. make sure that x2 is positive, if not, factor out the negative sign first. <br />Example J. Factor –x3 + 3x + 2x2<br />–x3 + 3x + 2x2 Arrange the terms in order<br />= –x3 + 2x2 + 3x Factor out the GCF<br />= – x(x2 – 2x – 3) <br />
80. 80. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />3. make sure that x2 is positive, if not, factor out the negative sign first. <br />Example J. Factor –x3 + 3x + 2x2<br />–x3 + 3x + 2x2 Arrange the terms in order<br />= –x3 + 2x2 + 3x Factor out the GCF<br />= – x(x2 – 2x – 3) <br />= – x(x – 3)(x + 1)<br />
81. 81. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order. <br />2. if there is any common factor, pull out the GCF first. <br />3. make sure that x2 is positive, if not, factor out the negative sign first. <br />Example J. Factor –x3 + 3x + 2x2<br />–x3 + 3x + 2x2 Arrange the terms in order<br />= –x3 + 2x2 + 3x Factor out the GCF<br />= – x(x2 – 2x – 3) <br />= – x(x – 3)(x + 1)<br />
82. 82. Factoring Trinomials II<br />Ex. A. Factor the following trinomials. If it’s prime, state so.<br />1. 3x2 – x – 2<br />2. 3x2 + x – 2<br />3. 3x2 – 2x – 1<br />4. 3x2 + 2x – 1<br />5. 2x2 – 3x + 1<br />6. 2x2 + 3x – 1<br />8. 2x2 – 3x – 2<br />7. 2x2 + 3x – 2<br />9. 5x2 – 3x – 2<br />12. 3x2 – 5x + 2<br />11. 3x2 + 5x + 2<br />10. 5x2 + 9x – 2<br />15. 6x2 + 5x – 6<br />14. 6x2 – 5x – 6<br />13. 3x2 – 5x + 2<br />16. 6x2 – x – 2<br />17. 6x2 – 13x + 2<br />18. 6x2 – 13x + 2<br />19. 6x2 + 7x + 2<br />20. 6x2 – 7x + 2 <br />21. 6x2 – 13x + 6<br />23. 6x2 – 5x – 4<br />24. 6x2 – 13x + 8<br />22. 6x2 + 13x + 6<br />25. 4x2 – 9<br />26. 4x2 – 49<br />25. 6x2 – 13x – 8<br />27. 25x2 – 4<br />28. 4x2 + 9<br />29. 25x2 + 9<br />B. Factor. Factor out the GCF, the “–”, and arrange the terms in order first.<br />32. –6x3 – x2 + 2x<br />30. – 6x2 – 5xy + 6y2<br />31. – 3x2 + 2x3– 2x<br />33. –15x2 – 25x2 – 10x<br />34. 12x2y2 –14x2y2 + 4xy2<br />