6.7
Factoring Trinomials:
The ac Method
6.7
OBJECTIVES
1. Use the ac test to determine whether a trinomial
is factorable over the integers
2. Use the results of the ac test to factor a trinomial
3. For a given value of x, evaluate f(x) before and
after factoring
The product of two binomials of the form
(__x __)(__x __)
will be a trinomial. In your earlier mathematics classes, you used the FOIL method to find
the product of two binomials. In this section, we will use the factoring by grouping method
to find the binomial factors for a trinomial.
First, let’s look at some factored trinomials.
Example 1
Matching Trinomials and Their Factors
Determine which of the following are true statements.
(a) x2 2x 8 (x 4)(x 2)
This is a true statement. Using the FOIL method, we see that
(x 4)(x 2) x2 2x 4x 8
x2 2x 8
(b) x2 6x 5 (x 2)(x 3)
Not a true statement, because
(x 2)(x 3) x2 3x 2x 6
x2 5x 6
(c) x2 5x 14 (x 2)(x 7)
True, because
(x 2)(x 7) x2 7x 2x 14
x2 5x 14
(d) x2 8x 15 (x 5)(x 3)
© 2001 McGraw-Hill Companies
False, because
(x 5)(x 3) x2 3x 5x 15
x2 8x 15
CHECK YOURSELF 1
Determine which of the following are true statements.
(a) 2x2 2x 3 (2x 3)(x 1)
(c) 2x2 7x 3 (x 3)(2x 1)
(b) 3x2 11x 4 (3x 1)(x 4)
431
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
The first step in learning to factor a trinomial is to identify its coefficients. To be consistent, we first write the trinomial in standard ax2 bx c form, then label the three coefficients as a, b, and c.
Example 2
Identifying the Coefficients of ax2 bx c
When necessary, rewrite the trinomial in ax2 bx c form. Then label a, b, and c.
(a) x2 3x 18
a1
b 3
c 18
(b) x2 24x 23
a1
b 24
c 23
(c) x2 8 11x
First rewrite the trinomial in descending order.
x 11x 8
2
Then,
a1
b 11
c8
CHECK YOURSELF 2
When necessary, rewrite the trinomial in ax2 bx c form. Then label a, b, and c.
(a) x2 5x 14
(b) x2 18x 17
(c) x 6 2x2
Not all trinomials can be factored. To discover if a trinomial is factorable, we try the ac test.
Rules and Properties:
The ac Test
A trinomial of the form ax2 bx c is factorable if (and only if) there are two
integers, m and n, such that
ac mn
and
bmn
In Example 3, we will determine whether each trinomial is factorable by finding the values
of m and n.
Example 3
Using the ac Test
Use the ac test to determine which of the following trinomials can be factored. Find the
values of m and n for each trinomial that can be factored.
(a) x2 3x 18
First, we note that a 1, b 3, and c 18, so ac 1(18) 18.
Then, we look for two numbers, m and n, such that mn ac and m n b. In this
case, that means
mn 18
m n 3
© 2001 McGraw-Hill Companies
432
FACTORING TRINOMIALS: THE ac METHOD
SECTION 6.7
433
We will look at every pair of integers with a product of 18. We then look at the sum of
each pair.
mn
mn
1(18) 18
2(9) 18
3(6) 18
1 (18) 17
2 (9) 7
3 (6) 3
We need look no further because we have found two integers whose mn product is 18 and
m n sum is 3.
m3
n 6
(b) x2 24x 23
We see that a 1, b 24, and c 23. So, ac 23 and b 24. Therefore, we want
mn 23
m n 24
We now work with integer pairs, looking for two integers with a product of 23 and a sum
of 24.
mn
mn
1(23)
23
1(23) 23
1 23
24
1 (23) 24
We find that m 1 and n 23.
(c) x2 11x 8
We see that a 1, b 11, and c 8. So, ac 8 and b 11. Therefore, we want
mn 8
m n 11
mn
mn
1(8)
8
2(4)
8
1(8) 8
2(4) 8
18
9
24
6
1 (8) 9
2 (4) 6
There is no other pair of integers with a product of 8, and none has a sum of 11. The
trinomial x2 11x 8 is not factorable.
(d) 2x2 7x 15
© 2001 McGraw-Hill Companies
We see that a 2, b 7, and c 15. So, ac 30 and b 7. Therefore, we want
mn 30
mn7
mn
mn
1(30) 30
2(15) 30
3(10) 30
5(6) 30
6(5) 30
10(3) 30
1 (30) 29
2 (15) 13
3 (10) 7
5 (6) 1
6 (5) 1
10 (3) 7
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
There is no need to go further. We have found two integers with a product of 30 and a sum
of 7. So m 10 and n 3.
In this example, you may have noticed patterns and shortcuts that make it easier to find
m and n. By all means, use those patterns. This is essential in mathematical thinking. You
are taught a step-by-step process that will always work for solving a problem; this process
is called an algorithm. It is very easy to teach a computer an algorithm. It is very difficult
(some would say impossible) for a computer to have insight. Shortcuts that you discover are
insights. They may be the most important part of your mathematical education.
CHECK YOURSELF 3
Use the ac test to determine which of the following trinomials can be factored. Find
the values of m and n for each trinomial that can be factored.
(a) x2 7x 12
(c) 2x2 x 6
(b) x2 5x 14
(d) 3x2 6x 7
So far we have used the results of the ac test only to determine whether a trinomial is
factorable. The results can also be used to help factor the trinomial.
Example 4
Using the Results of the ac Test to Factor a Trinomial
Rewrite the middle term as the sum of two terms, then factor by grouping.
(a) x2 3x 18
We see that a 1, b 3, and c 18, so
ac 18
b 3
We are looking for two numbers, m and n, so that
mn 18
m n 3
In Example 3, we found that the two integers were 3 and 6 because 3(6) 18 and
3 (6) 3. That result is used to rewrite the middle term (here 3x) as the sum of
two terms. We now rewrite the middle term as the sum of 3x and 6x.
x2 3x 6x 18
Then, we factor by grouping:
x2 3x 6x 18 x(x 3) 6(x 3)
(x 3)(x 6)
(b) x2 24x 23
We use the results of Example 3(b), in which we found m 1 and n 23, to rewrite
the middle term of the expression.
x2 24x 23 x2 x 23x 23
Then, we factor by grouping:
x2 x 23x 23 x(x 1) 23(x 1)
(x 1)(x 23)
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FACTORING TRINOMIALS: THE ac METHOD
SECTION 6.7
435
(c) 2x2 7x 15
From example 3(d ), we know that this trinomial is factorable and that m 10 and
n 3. We use that result to rewrite the middle term of the trinomial.
2x2 7x 15 2x2 10x 3x 15
2x(x 5) 3(x 5)
(x 5)(2x 3)
CHECK YOURSELF 4
Rewrite the middle term as the sum of two terms, then factor by grouping.
(a) x2 7x 12
(c) 2x2 x 6
(b) x2 5x 14
(d) 3x2 7x 6
Not all product pairs need to be tried to find m and n. A look at the sign pattern will eliminate many of the possibilities. Assuming the lead coefficient to be positive, there are four
possible sign patterns.
Pattern
Example
Conclusion
1. b and c are both positive.
2. b is negative and c is
2x 13x 15
x2 3x 2
m and n must be positive.
m and n must both be negative.
x2 5x 14
m and n are of opposite signs.
(The value with the larger
absolute value is positive.)
m and n are of opposite
signs. (The value with the
larger absolute value is
negative.)
2
positive.
3. b is positive and c is
negative.
© 2001 McGraw-Hill Companies
4. b and c are both negative.
x2 4x 4
Sometimes the factors of a trinomial seem obvious. At other times you might be certain
that there are only a couple of possible sets of factors for a trinomial. It is perfectly acceptable to check these proposed factors to see if they work. If you find the factors in this manner, we say that you have used the trial and error method. This method is discussed in
Section 6.7*, which follows this section.
To this point we have been factoring polynomial expressions. When a function is defined
by a polynomial expression, we can factor that expression without affecting any of the ordered pairs associated with the function. Factoring the expression makes it easier to find
some of the ordered pairs.
In particular, we will be looking for values of x that cause f(x) to be 0. We do this by
using the zero product rule.
Rules and Properties:
Zero Product Rule
If 0 ab, then either a 0, b 0, or both are zero.
Another way to say this is, if the product of two numbers is zero, then at least one of those
numbers must be zero.
CHAPTER 6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
Example 5
Factoring Polynomial Functions
Given the function f (x) 2x2 7x 15, complete the following.
(a) Rewrite the function in factored form.
From Example 4(c) we have
f (x) (x 5)(2x 3)
(b) Find the ordered pair associated with f (0).
f (0) (0 5)(0 3) 15
The ordered pair is (0, 15).
(c) Find all ordered pairs (x, 0).
We are looking for the x value for which f (x) 0, so
0 (x 5)(2x 3)
By the zero product rule, we know that either
(x 5) 0
(2x 3) 0
or
which means that
x 5
or
x
3
2
The ordered pairs are (5, 0) and
2, 0. Check the original function to see that these
3
ordered pairs are associated with that function.
CHECK YOURSELF 5
Given the function f(x) 2x2 x 6, complete the following.
(a) Rewrite the function in factored form.
(b) Find the ordered pair associated with f (0).
(c) Find all ordered pairs (x, 0).
CHECK YOURSELF ANSWERS
1. (a) False; (b) true; (c) true
2. (a) a 1, b 5, c 14; (b) a 1, b 18, c 17; (c) a 2, b 1, c 6
3. (a) Factorable, m 4, n 3; (b) factorable, m 7, n 2;
(c) factorable, m 4, n 3; (d) not factorable
4. (a) x2 3x 4x 12 (x 3)(x 4); (b) x2 7x 2x 14 (x 7)(x 2);
(c) 2x2 x 6 2x2 4x 3x 6 (2x 3)(x 2);
(d) 3x2 7x 6 3x2 9x 2x 6 (3x 2)(x 3)
3
5. (a) f (x) (2x 3)(x 2); (b) (0, 6); (c) , 0 and (2, 0)
2
© 2001 McGraw-Hill Companies
436
Name
Exercises
6.7
Section
Date
In exercises 1 to 8, determine which are true statements.
ANSWERS
1. x2 2x 3 (x 1)(x 3)
1.
2. x2 2x 8 (x 2)(x 4)
2.
3.
3. 2x2 5x 4 (2x 1)(x 4)
4.
5.
4. 3x2 13x 10 (3x 2)(x 5)
6.
5. x2 x 6 (x 5)(x 1)
7.
8.
6. 6x2 7x 3 (3x 1)(2x 3)
9.
7. 2x2 11x 5 (x 5)(2x 1)
10.
11.
8. 6x 13x 6 (2x 3)(3x 2)
2
12.
In exercises 9 to 16, when necessary, rewrite the trinomial in ax2 bx c form, then
label a, b, and c.
9. x2 3x 5
10. x2 2x 1
13.
14.
15.
16.
11. 2x 5x 3
2
12. 3x x 2
2
© 2001 McGraw-Hill Companies
17.
13. x 1 2x2
14. 4 5x 3x2
15. 2x 3x2 5
16. x x2 4
18.
19.
20.
In exercises 17 to 24, use the ac test to determine which trinomials can be factored. Find
the values of m and n for each trinomial that can be factored.
17. x2 3x 10
18. x2 x 12
19. x2 2x 3
20. 6x2 7x 2
437
ANSWERS
21.
21. 2x2 3x 2
22. 3x2 10x 8
23. 2x2 5x 2
24. 3x2 x 2
22.
23.
24.
In exercises 25 to 70, completely factor each polynomial expression.
25.
25. x2 7x 12
26. x2 9x 20
27. x2 9x 8
28. x2 11x 10
29. x2 15x 50
30. x2 13x 40
31. x2 7x 30
32. x2 7x 18
33. x2 10x 24
34. x2 13x 30
35. x2 7x 44
36. x2 15x 54
37. x2 8xy 15y2
38. x2 9xy 20y2
39. x2 16xy 55y2
40. x2 9xy 22y2
41. 3x2 11x 20
42. 2x2 9x 18
43. 5x2 18x 8
44. 3x2 20x 7
45. 12x2 23x 5
46. 8x2 30x 7
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
43.
44.
45.
46.
438
© 2001 McGraw-Hill Companies
42.
ANSWERS
47. 4x2 20x 25
47.
48. 9x2 24x 16
48.
49.
49. 5x 19x 30
50. 3x 17x 28
2
2
50.
51.
51. 5x2 24x 36
52.
52. 3x2 14x 24
53.
54.
53. 10x2 7x 12
54. 6x2 5x 21
55.
56.
55. 16x2 40x 25
56. 18x2 45x 7
57.
58.
57. 7x2 17xy 6y2
58. 5x2 17xy 12y2
59.
60.
61.
59. 8x 30xy 7y
2
2
60. 8x 14xy 15y
2
2
62.
63.
61. 3x2 24x 45
62. 2x2 10x 28
64.
65.
63. 2x2 26x 72
64. 3x2 39x 120
66.
67.
68.
© 2001 McGraw-Hill Companies
65. 6x3 31x2 5x
66. 8x3 25x2 3x
69.
70.
67. 5x3 14x2 24x
68. 3x4 17x3 28x2
69. 3x3 15x2y 18xy2
70. 2x3 10x2y 72xy2
439
ANSWERS
71.
(a)
(b)
(c)
In exercises 71 to 76, for each function, (a) rewrite the function in factored form, (b) find
the ordered pair associated with f (0), and (c) find all ordered pairs (x, 0).
72.
(a)
(b)
(c)
71. f (x) x2 2x 3
72. f (x) x2 3x 10
73.
(a)
(b)
73. f (x) 2x2 3x 2
74. f (x) 3x2 11x 6
75. f (x) 3x2 5x 28
76. f (x) 10x2 13x 3
(c)
74.
(a)
(b)
(c)
75.
(a)
(b)
(c)
76.
(a)
(b)
Certain trinomials in quadratic form can be factored with similar techniques. For
instance, we can factor x4 5x2 6 as (x2 6)(x2 1). In exercises 77 to 88, apply a
similar method to completely factor each polynomial.
(c)
77. x4 3x2 2
78. x4 7x2 10
79. x4 8x2 33
80. x4 5x2 14
81. y6 2y3 15
82. x6 10x3 21
83. x5 6x3 16x
84. x6 8x4 15x2
85. x4 5x2 36
86. x4 5x2 4
87. x6 6x3 16
88. x6 2x3 3
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
88.
89.
90.
In exercises 89 to 96, determine a value of the number k so that the polynomial can be
factored.
89. x2 5x k
440
90. x2 3x k
© 2001 McGraw-Hill Companies
87.
ANSWERS
91. 6x2 x k
91.
92. 4x2 x k
92.
93.
93. x kx 6
94. x kx 15
2
2
94.
95.
95. 6x2 kx 3
96. 2x2 kx 15
96.
97.
97. The product of three numbers is x 6x 8x. Show that the numbers are
3
2
consecutive even integers. (Hint: Factor the expression.)
98.
99.
(a)
(b)
100. (a)
98. The product of three numbers is x3 3x2 2x. Show that the numbers are
consecutive integers.
(b)
101. (a)
(b)
102. (a)
In each of the following, (a) factor the given function, (b) identify the values of x for
which f (x) 0, (c) graph f(x) using the graphing calculator and determine where the
graph crosses the x axis, and (d) compare the results of (b) and (c).
(b)
103.
99. f (x) x2 2x 8
100. f (x) x2 3x 10
101. f (x) 2x2 x 3
102. f (x) 3x2 x 2
104.
In exercises 103 and 104, determine the binomials that represent the dimensions of the
given figure.
103.
104.
© 2001 McGraw-Hill Companies
Area 2x2 7x 15
?
?
Area 3x2 11x 10
?
?
441
Answers
1. True
3. False
5. False
7. False
9. a 1, b 3, c 5
11. a 2, b 5, c 3
13. a 2, b 1, c 1
15. a 3, b 2, c 5
17. Factorable, m 5, n 2
19. Not factorable
21. Not factorable
23. Factorable, m 4, n 1
25. (x 3)(x 4)
27. (x 8)(x 1)
29. (x 10)(x 5)
31. (x 10)(x 3)
33. (x 6)(x 4)
35. (x 11)(x 4)
37. (x 3y)(x 5y)
39. (x 11y)(x 5y)
41. (3x 4)(x 5)
43. (5x 2)(x 4)
45. (3x 5)(4x 1)
47. (2x 5)2
49. (5x 6)(x 5)
51. (5x 6)(x 6)
53. (2x 3)(5x 4)
55. (4x 5)2
57. (7x 3y)(x 2y)
59. (4x y)(2x 7y)
61. 3(x 5)(x 3)
63. 2(x 4)(x 9)
65. x(6x 1)(x 5)
67. x(x 4)(5x 6)
69. 3x(x 6y)(x y)
71. (a) (x 3)(x 1); (b) (0, 3); (c) (3, 0) and (1, 0)
2, 0 and (2, 0)
7
75. (a) (x 4)(3x 7); (b) (0, 28); (c) (4, 0) and , 0
3
73. (a) (2x 1)(x 2); (b) (0, 2); (c)
1
(x2 1)(x2 2)
79. (x2 11)(x2 3)
81. (y3 5)(y3 3)
2
2
2
x(x 2)(x 8)
85. (x 3)(x 3)(x 4)
(x 2)(x2 2x 4)(x3 2)
89. 4
91. 2, 1, or 5
5, 5, 1, or 1
95. 7, 7, 17, 17, 3, or 3
97. x(x 2)(x 4)
3
99. (a) (x 4)(x 2); (b) 4, 2
101. (a) (2x 3)(x 1); (b) , 1
2
103. (2x 3) and (x 5)
© 2001 McGraw-Hill Companies
77.
83.
87.
93.
442