2. Factoring Trinomials II<br />Now let’s try to factor trinomials of the form ax2 + bx + c.<br />
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 />
18. Factoring Trinomials II<br />(Reversed FOIL Method) <br />Let’s see how the above examples are related to factoring. <br />
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. 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. 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. 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. 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. 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. 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. 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. 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 />
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. 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. 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. 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. 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. 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. 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. 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. 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 />
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 />
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. 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. 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. 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. 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. 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. 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. 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. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />
74. Factoring Trinomials II<br />Finally, before starting the reverse-FOIL procedure<br />1. make sure the terms are arranged in order.<br />
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. 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. 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. 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. 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. 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. 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 />
Be the first to comment