Samples from Basic One Step Ch2

In the factoring of trinomials of the previous sections, it may have been assumed that the coefficient of x² is 1 (or that the coefficient is a common factor of the entire trinomial). Notice that as you work through this section, because of combinations of numbers, the trial and error process becomes more and more challenging. However, let=s begin with some examples that do not involve too many combinations of numbers.

EXAMPLE 1. 5x² + 6x + 1.

Solution: Remember that this should be factored by F L OI.

The F term is 5x², L is 1,

5x² + 6x + 1 5x² + 6x + 1

(5x )(x ) ( + 1)( + 1)

and OI adds up to 6x, as follows:

5x² + 6x + 1

Final answer: (5x + 1)(x + 1)

EXAMPLE 2. 3x² + 4x + 1

Solution: The F term is 3x², L is 1,

3x² + 4x + 1 3x² + 4x + 1

(3x )(x ) ( + 1)( + 1)

and OI adds up to 4x, as follows:

3x² + 4x + 1

Final answer: (3x + 1)(x + 1)

EXAMPLE 3. 8x² + 9x + 1 (Trial and Error!)

Solution: The F term is 8x², L is 1,

8x² + 9x + 1 8x² + 9x + 1

(8x )(x ) or (4x )(2x ) ( + 1)( + 1)

Since the OI term must add up to 9x, use the 8x, 1x combination.

8x² + 9x + 1

Final answer: (8x + 1)(x + 1)

EXAMPLE 4. 8x² − 6x + 1 (Trial and Error!)

Solution: The F term is 8x², L is 1,

8x² − 6x + 1 8x² − 6x + 1

(8x )(x ) or (4x )(2x ) ( − 1)( − 1)

Since the OI term must add up to −6x, use the 4x, 2x combination.

8x² − 6x + 1

Final answer: (4x − 1)(2x − 1)

EXERCISES. Factor each of the following trinomials.

1. 3x² + 4x + 1 2. 4x² + 5x + 1 3. 7x² + 8x + 1

( )( ) ( )( ) ( )( )

4. 3x² − 4x + 1 5. 4x² − 5x + 1 6. 7x² − 8x + 1

( )( ) ( )( ) ( )( )

Text Box:
REMEMBER: When the L term is positive, add the O and I terms.
When the L term is negative, subtract the O and I terms.

EXAMPLE 5. 8x² − 2x − 1 (Trial and Error!)

NOTE: Since the L term is negative, you must subtract the O and I terms.

Solution: The F term is 8x², L is −1,

8x² − 2x − 1 8x² − 2x − 1

(8x )(x ) or (4x )(2x ) ( + 1)( − 1) or ( − 1)( + 1)

Since the OI term must subtract to give −2x, use the −4x, 2x combination.

8x² − 2x − 1

Final answer: (4x + 1)(2x − 1)

EXERCISES. Factor each of the following trinomials.

7. 3x² + 2x − 1 8. 4x² + 3x − 1 9. 7x² + 6x − 1

( )( ) ( )( ) ( )( )

10. 3x² − 2x − 1 11. 4x² − 3x − 1 12. 7x² − 6x − 1

( )( ) ( )( ) ( )( )

13. 6x² + 5x + 1 14. 6x² + 7x + 1 15. 6x² − 7x + 1

16. 6x² − 5x + 1 17. 6x² + 5x − 1 18. 6x² − 5x − 1

19. 8x² − 9x + 1 20. 8x² + 7x − 1 21. 8x² + 2x − 1

22. 8x² + 6x + 1 23. 10x² + 7x + 1 24. 10x² − 11x + 1

25. 10x² − 3x − 1 26. 10x² + 9x − 1 27. 10x² + 11x + 1

28. 10x² − 7x + 1 29. 10x² − 9x − 1 30. 10x² + 3x − 1

Of course, with larger numbers, with many more combinations of numbers this can become a very lengthy process of trial and error. There are some systematic methods of factoring these trinomials, which usually turn out to be somewhat complicated. In problems that are not too difficult, the trial and error method will be fairly simple and more than adequate for now.

EXAMPLE 6. 5x² + 8x + 3 (Trial and Error!!!)

Solution: The F term is 5x², L is 3,

5x² + 8x + 3 5x² + 8x + 3

(5x )(x ) ( + 3)( + 1) or ( + 1)( + 3)

The OI term must add up to 8x.

5x² + 8x + 3

(5x + 3)(x + 1) or (5x + 1)(x + 3) (The first is correct, the second is NOT!)

Final answer: (5x + 3)((x + 1)

EXAMPLE 7. 5x² + 16x + 3 (Trial and Error!)

Solution: The F term is 5x², L is 3,

5x² + 16x + 3 5x² + 16x + 3

(5x )(x ) ( + 3)( + 1) or ( + 1)( + 3)

The OI term must add up to 16x.

5x² + 16x + 3

(5x + 3)(x + 1) or (5x + 1)(x + 3) (The second is correct, NOT the first!)

Final answer: (5x + 1)((x + 3)

EXAMPLE 8. EXAMPLE 9.

3x² + 10x + 7 3x² + 22x + 7

( )( ) ( )( )

(3x )( x ) (3x )( x )

In Examples 8 and 9, the F term must obviously be 3x @ x. The L term is 7, which must be either

7 @ 1 or 1 @ 7 . The possibilities are

(3x + 7)(x + 1) whose middle term is 3x + 7x = 10x,

or (3x + 1)(x + 7) whose middle term is 21x + 1x = 22x.

Final answers: Example 8. 3x² + 10x + 7 = (3x + 7)(x + 1)

Example 9. 3x² + 22x + 7 = (3x + 1)(x + 7)

In Examples 10 and 11, the L term is negative, so you must subtract the O and I terms.

EXAMPLE 10. EXAMPLE 11.

3x² + 4x − 7 3x² − 20x − 7

( )( ) ( )( )

(3x )( x ) (3x )( x )

In Examples 9 and 10, the F term again is obviously 3x @ x, and the L term is −7, which means −7 @ 1 or 7 @ −1 (opposite signs!). The possibilities are:

(3x−7)(x+1) whose middle term is 3x−7x = −4x,

(3x+7)(x−1) whose middle term is −3x+7x = 4x,

(3x−1)(x+7) whose middle term is 21x−1x = 20x,

or (3x+1)(x−7) whose middle term is −21x+1x = −20x

Final answers: Example 10. 3x² + 4x − 7 = (3x + 7)(x − 1)

Example 11. 3x² − 20x − 7 = (3x + 1)(x − 7)

EXERCISES. Factor each of the following trinomials.

31. 3x² + 8x + 5 32. 3x² + 16x + 5 33. 3x² + 2x − 5

34. 3x² − 14x − 5 35. 3x² − 14x + 11 36. 3x² + 8x − 11

37. 3x² − 8x − 11 38. 3x² + 34x + 11 39. 3x² − 34x + 11

40. 3x² + 32x − 11 41. 5x² + 6x − 11 42. 5x² + 54x − 11

43. 5x² + 41x + 8 44. 5x² + 14x + 8 45. 5x² + 22x + 8

46. 5x² + 13x + 8 47. 5x² − 14x + 8 48. 5x² − 41x + 8

49. 5x² + 3x − 8 50. 5x² + 39x − 8 51. 5x² + 6x − 8

52. 5x² − 6x − 8 53. 5x² + 18x − 8 54. 5x² − 13x + 8

55. 5x² − 31x + 6 56. 5x² − 11x + 6 57. 5x² + 13x + 6

58. 5x² + 17x + 6 59. 5x² − 13x − 6 60. 5x² − 7x − 6

61. 5x² − x − 6 62. 5x² + 29x − 6 63. 6x² + 49x + 8

64. 6x² − 49x + 8 65. 6x² + 47x − 8 66. 6x² + 19x + 8

67. 6x² − 19x + 8 68. 6x² − 13x − 8 69. 6x² + 13x − 8

70. 6x² + 19x + 10 71. 6x² + 11x − 10 72. 6x² + 17x + 10

ANSWERS 2.05

p.166-172: (NOTE: Factors may be given in any order!)

1. (3x+1)(x+1); 2. (4x+1)(x+1); 3. (7x+1)(x+1); 4. (3x-1)(x-1); 5. (4x-1)(x-1);

6. (7x-1)(x-1); 7. (3x-1)(x+1); 8. (4x-1)(x+1); 9. (7x-1)(x+1); 10. (3x+1)(x-1);