Do you remember
from Section 2.01, you learned to multiply the product of two binomials
using the F OI L method? As an introduction to this most important
section on factoring trinomials, it will be helpful to review the
F OI L method of finding products.
REVIEW
EXERCISES. Use the F OI L method to multiply each of the
following:
1. (x +
2)(x + 5) 2. (x + 3)(x +
2) 3. (x + 5)(x + 3)
4. (x
− 2)(x −
5) 5. (x − 3)(x − 2)
6. (x − 5)(x − 3)
7. (x
− 2)(x +
5) 8. (x + 3)(x − 2)
9. (x + 5)(x − 3)
10. (x +
2)(x − 5)
11. (x − 3)(x + 2)
12. (x − 5)(x + 3)
In each of the
previous exercises, you were given a product of two binomials, and with
F OI L, in each case you obtained a trinomial with x2
as the first term. Now the problem will be to work these problems in
reverse. What if you were given a trinomial, such as x2+7x+10
and asked to factor it−−that is, to express it as a product. This
product is the product of two binomials. When you factor the
trinomial: x2 + 7x + 10 you expect the
product of binomials: ( )( ).
Also, when
factoring a trinomial, instead of thinking F OI L, you need to change
the order and think F L OI. In other words, you need to find the
correct F (first times first) combination, then skip to the L (last
times last). Finally, check to make sure the OI (outer times outer,
inner times inner) middle term is correct. F OI L
EXAMPLE 1. Factor
x2 + 7x + 10.
Solution: x2 + 7x + 10
(
)( ) Product of two binomials;
(x )(x
) F term is x2, which is x times x;
(x )(x
) L term is +10.
You must find two
numbers whose product is +10. Probably 2 times 5, or it could be 1
times 10. Try 2 times 5. Since the last sign is positive, it will be
positive times positive, or negative times negative. Also, the middle
terms (O and I) must be added together.
(x + 2)(x
+ 5) OI term is 7x. This means the outer times outer and the
inner times inner terms must add up to 7x.
[NOTICE: The
order does not matter! If you wrote (x + 5)(x + 2), this
is exactly equivalent to (x + 2)(x + 5). WHY??]
EXERCISES:
Factor. (Find two numbers whose product is "L," and whose sum is "OI."
)
1. x2
+ 5x + 6
2. x2 + 8x + 15
3. x2 + 5x + 4
(x
+____)(x +____)
(x +____)(x +____)
(x +____)(x +____)
4. x2
+ 7x + 6
5. x2 + 6x + 8
6.
x2 + 9x + 8
(x
+____)(x +____)
(x +____)(x +____)
(x +____)(x +____)
7. x2
+ 3x + 2
8. x2 + 9x + 14
9. x2 + 7x + 10
(x
+____)(x +____)
(x +____)(x +____)
(x +____)(x +____)
10. x2
+ 12x + 35
11. x2 + 13x + 22
12. x2 + 9x + 18
(x
+____)(x +____)
(x +____)(x +____)
(x +____)(x +____)
13. x2
+ 13x + 40 14.
x2 + 14x + 40
15. x2 + 41x + 40
(x
)(x )
(x )(x )
(x )(x )
16. x2
+ 17x + 16 17.
x2 + 10x + 16
18. x2 + 17x + 72
(x
)(x ) (x
)(x )
(x )(x )
In the next
exercises, notice that the sign of the middle term is negative while the
sign of the last term is positive. This means that the signs will be
like sign, both negative.
EXAMPLE 2. Factor
x2 − 7x + 10.
Solution: x2 − 7x + 10
This begins exactly as Example 1.
(
)( ) Product of two binomials;
(x
)(x ) F term is x2, which is x
times x;
(x )(x
) L term is +10.
You must find two
numbers whose product is +10. Probably 2 times 5, or it could be 1
times 10. Try 2 times 5. Since the last sign is positive, it will be
positive times positive, or negative times negative. Also, the middle
terms (O and I) must be added together.
(x
− 2)(x − 5) OI term
is − 7x. This means the outer times outer and the inner
times inner terms must add up to −7x.
19. x2
− 5x + 6 20. x2
− 8x + 15 21. x2
− 7x + 10
(x
)(x ) (x )(x
) (x )(x )
22. x2
− 9x + 14 23. x2
− 6x + 8 24. x2
− 7x + 12
25. x2
− 15x + 14 26. x2
− 9x + 8 27. x2
− 13x + 42
28. x2
− 15x + 54 29. x2
− 13x + 36 30. x2
− 20x + 36
If the "L" term is
negative, this means that the signs are opposite, and you must subtract
the numbers to get the middle term:
F OI L
F OI L
EXAMPLE 3.
x2 + 2x
− 8 EXAMPLE
4. x2 − 2x − 8
Solutions:
Because the L term is negative in each of these examples, you
must have opposite signs, probably −2 times 4 or −4 times 2.
x2 + 2x − 8
x2 − 2x −
8
(x + )(x − )
(x + )(x − )
Answers:
(x + 4)(x − 2)
(x + 2)(x − 4)
[Also
correct: (x − 2)(x + 4)
(x − 4)(x + 2)
]
31. x2
+ 2x − 3
32. x2 + 3x − 4
33. x2 + x − 6
34. x2
+ x − 2
35. x2 + 3x − 10
36. x2 + 5x − 14
37. x2
− 4x − 5
38. x2 − 2x − 3
39. x2 − 5x − 14
40. x2
− 7x − 18
41. x2 + 3x
− 18 42.
x2 − x − 20
43. x2
+ x − 20
44. x2 + 2x − 35
45. x2 − 9x − 22
46. x2
− 6x − 27
47. x2 + 2x − 24
48. x2 − 5x − 24
49. x2
− 2x − 24
50. x2 + 2x − 24
51. x2 + 10x − 24
52. x2
− 7x − 30
53. x2 − x − 30
54. x2 − 13x − 30
Perhaps you have
noticed that the key to factoring trinomials is breaking down the LAST
term. It may be helpful to list all possible combinations of factors.
To do this, begin with 1 times the number, then try 2, 3, 4, 5, etc.
Notice that as the first number gets larger, the second number gets
smaller, and the numbers "meet" in the middle. When the numbers meet,
you have all the combinations. Here are some examples:
10 12
36 60
1 X
10 1 X 12 1 X 36
1 X 60
2 X
5 2 X 6 2 X
18 2 X 30
3 X 4 3 X
12 3 X 20
4 X 9
4 X 15
6 X 6
5 X 12
6 X 10
It is also worth
noting that not all trinomials can be factored. The trinomials x2
+ x + 2, x2 + 2x + 6, and x2
− 4x − 6 are examples of trinomials that cannot be
factored. These are called prime trinomials.
The following is a
summary of trinomial (F L OI)
factoring.
SUMMARY
RULES: 1. When the sign of the LAST is positive, the
signs are the SAME.
You find middle term by ADDING the O and I terms.
2. When the sign of the LAST is negative, the signs are
OPPOSITE.
You find middle term by SUBTRACTING the O and I terms.
|
EXERCISES.
Factor each of the following trinomials.
55. x2
− 7x + 12 56. x2 −
x − 12 57. x2 + 13x
+ 12
58. x2
− 8x + 12 59. x2 − 4x
− 12 60. x2 − 13x
+ 12
61. x2
− 13x + 36 62. x2 − 20x
+ 36 63. x2 + 16x −
36
64. x2
− 5x − 36 65. x2 − 35x
− 36 66. x2 − 12x
+ 36
67. x2
+ 17x + 60 68. x2 + 16x
+ 60 69. x2 + 19x
+ 60
70. x2
− 4x − 60 71. x2 + 7x
− 60 72. x2 − 28x
− 60
73. x2
− 32x + 60 74. x2 + 11x
− 60 75. x2 + 61x
+ 60
76. x2
− 23x + 60 77. x2 − 17x
− 60 78. x2 − 59x
− 60
79. x2
− x − 56 80. x2 −
x − 72 81. x2 + 17x
+ 72
82. x2
− 11x + 28 83. x2 − 3x
− 28 84. x2 + 16x
+ 28
85. x2
+ 13x + 42 86. x2 − x
− 42 87. x2 − 23x
+ 42
88. x2
− 5x − 50 89. x2 + 23x
− 50 90. x2 − 21x
+ 38
Frequently, it is
necessary to FACTOR THE COMMON FACTOR FIRST (FCFF). When there
is a common factor in the problem, always remember to FCFF! (NOTE:
These exercises require TWO steps−−the
AFactoring
Two Step!@)
EXERCISES.
Factor the trinomials completely. Be sure to FCFF!!
91. 3x2
+ 6x − 9 92. 5x2 + 15x
− 20 93. 8x2 + 8x −
48
3(x2
+ 2x − 3)
5( )
____( )
3( )( ) 5(
)( ) ____(
)( )
94. 6x2
+ 6x − 12 95. 12x2 + 36x
− 120 96. 10x2 + 30x −
100
____( )
____( )( )
97. x3
− 4x2 − 5x 98. x3
+ 5x2 + 6x
99. 2x3 − 14x2 + 20x
100. 5x3
+ 5x2 − 10x 101. 7x3
+ 49x2 + 42x 102. 8x3
− 40x2 + 32x
103. x4
+ x3 − 20x2
104. x4 + 2x3
− 35x2
105. 15x4
− 45x3 − 60x2
106. 20x4 − 20x3 − 120x2
107. 30x4
+ 90x3 − 300x2
108. 18x4 + 54x3
+ 36x2
ANSWERS 2.04
p. 147:
1. x2+7x+10;
2. x2+5x+6; 3. x2+8x+15;
4. x2-7x+10; 5. x2-5x+6;
6. x2-8x+15;
7. x2+3x-10;
8. x2+ x- 6; 9. x2+2x-15;
10. x2-3x-10; 11. x2- x-
6; 12. x2-2x-15
p.
149-156: (NOTE: Factors may be given
in any order!)
1. (x+3)(x+2); 2. (x+5)(x+3);
3. (x+4)(x+1); 4. (x+6)(x+1); 5. (x+4)(x+2);
6. (x+8)(x+1);
7. (x+2)(x+1); 8. (x+7)(x+2); 9. (x+5)(x+2);
10. (x+7)(x+5);
11. (x+11)(x+2);
12. (x+6)(x+3); 13. (x+5)(x+8); 14. (x+4)(x+10);
15. (x+40)(x+1);
16. (x+16)(x+1);
17. (x+8)(x+2); 18. (x+9)(x+8); 19. (x-3)(x-2);
20. (x-5)(x-3);
21. (x-5)(x-2);
22. (x-7)(x-2); 23. (x-4)(x-2); 24. (x-3)(x-4);
25. (x-14)(x-1);
26. (x-8)(x-1);
27. (x-7)(x-6); 28. (x-9)(x-6); 29. (x-9)(x-4);
30. (x-18)(x-2);
31. (x+3)(x-1); 32. (x+4)(x-1); 33. (x+3)(x-2);
34. (x+2)(x-1); 35. (x+5)(x-2);
36. (x+7)(x-2);
37. (x-5)(x+1); 38. (x-3)(x+1); 39. (x-7)(x+2);
40. (x-9)(x+2);
41. (x+6)(x-3);
42. (x-5)(x+4); 43. (x+5)(x-4); 44. (x+7)(x-5);
45. (x-11)(x+2);
46. (x-9)(x+3); 47. (x+6)(x-4); 48. (x-8)(x+3);
49. (x-6)(x+4); 50. (x+6)(x-4);
51. (x+12)(x-2);
52. (x-10)(x+3); 53. (x-6)(x+5); 54. (x-15)(x+2);
55. (x-4)(x-3);
56. (x-4)(x+3);
57. (x+12)(x+1); 58. (x-6)(x-2); 59. (x-6)(x+2);
60. (x-12)(x-1);
61. (x-9)(x-4); 62. (x-18)(x-2); 63. (x+18)(x-2);
64. (x-9)(x+4); 65. (x-36)(x+1);
66. (x-6)(x-6);
67. (x+5)(x+12); 68. (x+6)(x+10); 69. (x+15)(x+4);
70. (x-10)(x+6);
71. (x+12)(x-5);
72. (x-30)(x+2); 73. (x-30)(x-2); 74. (x+15)(x-4);
75. (x+60)(x+1);
76. (x-20)(x-3); 77. (x-20)(x+3); 78. (x-60)(x+1);
79. (x-8)(x+7); 80. (x-9)(x+8);
81. (x+9)(x+8);
82. (x-7)(x-4); 83. (x-7)(x+4); 84. (x+14)(x+2);
85. (x+7)(x+6); 86. (x-7)(x+6); 87. (x-21)(x-2);
88. (x-10)(x+5); 89. (x+25)(x-2);
90. (x-19)(x-2); 91. 3(x+3)(x-1); 92. 5(x+4)(x-1);
93. 8(x+3)(x-2); 94. 6(x+2)(x-1);
95. 12(x+5)(x-2); 96. 10(x+5)(x-2); 97.
x(x-5)(x+1); 98. x(x+2)(x+3);
99. 2x(x-5)(x-2);
100. 5x(x+2)(x-1); 101. 7x(x+6)(x+1);
102. 8x(x-4)(x-1); 103. x2(x+5)(x-4);
104. x2(x+7)(x-5);
105. 15x2(x-4)(x+1); 106. 20x2(x-3)(x+2);
107. 30x2(x+5)(x-2); 108. 18x2(x+2)(x+1).
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