Download 5.3 Factoring Polynomials of the Form ax 2 + bx + c

Section 5.3 / Factoring Polynomials of the Form ax 2 bx c
5.3
Objective A
251
Factoring Polynomials of the
Form ax 2 bx c
To factor a trinomial of the form
ax 2 bx c by using trial factors
Trinomials of the form ax2 bx c,
where a, b, and c are integers, are
shown at the right.
3x2 2x 4; a 3, b 1, c 4
6x2 2x 3; a 6, b 2, c 3
These trinomials differ from those in the preceding section in that the coefficient
of x2 is not 1. There are various methods of factoring these trinomials. The
method described in this objective is factoring polynomials using trial factors.
To reduce the number of trial factors that must be considered, remember the
following:
1. Use the signs of the constant term and the coefficient of x in the trinomial
to determine the signs of the binomial factors. If the constant term is positive, the signs of the binomial factors will be the same as the sign of the coefficient of x in the trinomial. If the sign of the constant term is negative, the
constant terms in the binomials have opposite signs.
2. If the terms of the trinomial do not have a common factor, then the terms of
neither of the binomial factors will have a common factor.
HOW TO
Video
Factor: 2x2 7x 3
The terms have no common
factor. The constant term is
positive. The coefficient of x
is negative. The binomial
constants will be negative.
Write trial factors. Use the
Outer and Inner products of
FOIL to determine the middle
term, 7x, of the trinomial.
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Write the factors of the trinomial.
HOW TO
Positive
Factors of 2
(coefficient of x 2)
1, 2
Negative
Factors of 3
(constant term)
1, 3
Trial Factors
Middle Term
x 1 2x 3
x 3 2x 1
3x 2x 5x
x 6x 7x
2x2 7x 3 x 32x 1
Factor: 3x2 14x 15
The terms have no common
factor. The constant term is
positive. The coefficient of x
is positive. The binomial
constants will be positive.
Write trial factors. Use the
Outer and Inner products of
FOIL to determine the middle
term, 14x, of the trinomial.
Write the factors of the trinomial.
Positive
Factors of 3
(coefficient of x 2)
Positive
Factors of 15
(constant term)
1, 3
1, 15
3, 5
Trial Factors
Middle Term
x 1 3x 15
x 15 3x 1
x 3 3x 5
x 5 3x 3
Common factor
x 45x 46x
5x 9x 14x
Common factor
3x2 14x 15 x 33x 5
252
Chapter 5 / Factoring
HOW TO
Factor: 6x3 14x2 12x
6x3 14x2 12x 2x3x2 7x 6
Factor the GCF, 2x, from the
terms.
Positive
Factors of 3
Factor the trinomial. The
constant term is negative.
The binomial constants will
have opposite signs.
Factors of 6
1, 6
1, 6
2, 3
2, 3
1, 3
Write trial factors. Use the
Outer and Inner products of
FOIL to determine the middle
term, 7x, of the trinomial.
Trial Factors
x 1 3x 6
x 6 3x 1
x 1 3x 6
x 6 3x 1
x 2 3x 3
x 3 3x 2
x 2 3x 3
x 3 3x 2
It is not necessary to test trial
factors that have a common
factor.
Write the factors of the trinomial.
Middle Term
Common factor
x 18x 17x
Common factor
x 18x 17x
Common factor
2x 9x 7x
Common factor
2x 9x 7x
6x3 14x2 12x 2xx 33x 2
For this example, all the trial factors were listed. Once the correct factors have
been found, however, the remaining trial factors can be omitted. For the examples and solutions in this text, all trial factors except those that have a
common factor will be listed.
Example 1
Factor: 3x2 x 2
Solution
You Try It 1
Factor: 2x2 x 3
Your solution
Positive
factors of 3: 1, 3
Factors of 2: 1, 2
1, 2
Trial Factors
Middle Term
x 1 3x 2
x 2 3x 1
x 1 3x 2
x 2 3x 1
2x 3x x
x 6x 5x
2x 3x x
x 6x 5x
Example 2
Factor: 12x3 32x2 12x
Solution
You Try It 2
Factor: 45y3 12y2 12y
Your solution
The GCF is 4x.
12x3 32x2 12x 4x3x2 8x 3
Factor the trinomial.
Positive
Factors of 3: 1, 3
1, 3
factors of 3: 1, 3
Trial Factors
Middle Term
x 3 3x 1
x 3 3x 1
x 9x 8x
x 9x 8x
12x3 32x2 12x 4xx 33x 1
Solutions on pp. S12– S13
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3x2 x 2 x 13x 2
Section 5.3 / Factoring Polynomials of the Form ax 2 bx c
Objective B
253
To factor a trinomial of the
form ax 2 bx c by grouping
In the preceding objective, trinomials of the form ax2 bx c were factored
by using trial factors. In this objective, these trinomials will be factored by
grouping.
To factor ax2 bx c, first find two factors of a c whose sum is b. Then use
factoring by grouping to write the factorization of the trinomial.
HOW TO
Factor: 2x2 13x 15
Find two positive factors of 30 2 15 whose sum is 13.
Positive Factors of 30
1, 30
2, 15
3, 10
Sum
31
17
13
• Once the required sum has been
found, the remaining factors need
not be checked.
2x2 13x 15 2x2 3x 10x 15
2x2 3x 10x 15
x2x 3 52x 3
2x 3x 5
• Use the factors of 30 whose sum is
13 to write 13x as 3x 10x .
• Factor by grouping.
Check: 2x 3x 5 2x2 10x 3x 15
2x2 13x 15
HOW TO
Factor: 6x2 11x 10
Find two factors of 60 610 whose sum is 11.
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Factors of 60
1, 60
1, 60
2, 30
2, 30
3, 20
3, 20
4, 15
Sum
59
59
28
28
17
17
11
6x2 11x 10 6x2 4x 15x 10
6x2 4x 15x 10
2x3x 2 53x 2
3x 22x 5
Check: 3x 22x 5 6x 2 15x 4x 10
6x 2 11x 10
• Use the factors of 60 whose
sum is 11 to write 11x as
4x 15x .
• Factor by grouping. Recall that
15x 10 (15x 10).
254
Chapter 5 / Factoring
HOW TO
Factor: 3x2 2x 4
Find two factors of 12 34 whose sum is 2.
Factors of 12
1, 12
1, 12
2, 6
2, 6
3, 4
3, 4
TA K E N O T E
3x 2x 4 is a prime
polynomial.
Sum
11
11
4
4
1
1
2
Because no integer factors of 12 have a sum of 2, 3x2 2x 4 is
nonfactorable over the integers.
Example 3
You Try It 3
Factor: 2x 19x 10
Factor: 2a2 13a 7
Solution
Your solution
2
Factors of 20 [2(10)]
Sum
1, 20
19
2x2 19x 10 2x2 x 20x 10
2x2 x 20x 10
x2x 1 102x 1
2x 1x 10
Example 4
You Try It 4
Factor: 24x y 76xy 40y
Factor: 15x3 40x2 80x
Solution
Your solution
2
Negative
Factors of 60 [6(10)]
Sum
1, 60
2, 30
3, 20
4, 15
61
32
23
19
6x2 19x 10 6x2 4x 15x 10
6x2 4x 15x 10
2x3x 2 53x 2
3x 22x 5
24x2y 76xy 40y 4y6x2 19x 10
4y3x 22x 5
Solutions on p. S13
Copyright © Houghton Mifflin Company. All rights reserved.
The GCF is 4y.
24x2y 76xy 40y 4y6x2 19x 10
Section 5.3 / Factoring Polynomials of the Form ax 2 bx c
255
5.3 Exercises
Objective A
To factor a trinomial of the form ax 2 bx c
by using trial factors
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For Exercises 1 to 70, factor by using trial factors.
1.
2x2 3x 1
2. 5x2 6x 1
3. 2y2 7y 3
4. 3y2 7y 2
5.
2a2 3a 1
6. 3a2 4a 1
7. 2b2 11b 5
8. 3b2 13b 4
9.
2x2 x 1
10. 4x2 3x 1
11. 2x2 5x 3
12. 3x2 5x 2
13. 2t2 t 10
14.
2t2 5t 12
15.
3p2 16p 5
16.
6p2 5p 1
17. 12y2 7y 1
18.
6y2 5y 1
19.
6z2 7z 3
20.
9z2 3z 2
21. 6t2 11t 4
22.
10t2 11t 3
23.
8x2 33x 4
24.
7x2 50x 7
25. 5x2 62x 7
26.
9x2 13x 4
27.
12y2 19y 5
28.
5y2 22y 8
29. 7a2 47a 14
30.
11a2 54a 5
31.
3b2 16b 16
32.
6b2 19b 15
33. 2z2 27z 14
34.
4z2 5z 6
35.
3p2 22p 16
36.
7p2 19p 10
Chapter 5 / Factoring
37. 4x2 6x 2
38.
12x2 33x 9
39.
15y2 50y 35
40.
30y2 10y 20
41. 2x3 11x2 5x
42.
2x3 3x2 5x
43.
3a2b 16ab 16b
44.
2a2b ab 21b
45. 3z2 95z 10
46.
8z2 36z 1
47.
36x 3x2 3x3
48.
2x3 2x2 4x
49. 80y2 36y 4
50.
24y2 24y 18
51.
8z3 14z2 3z
52.
6z3 23z2 20z
53. 6x2y 11xy 10y
54.
8x2y 27xy 9y
55.
10t2 5t 50
56. 16t2 40t 96
57.
3p3 16p2 5p
58.
6p3 5p2 p
59. 26z2 98z 24
60.
30z2 87z 30
61.
10y3 44y2 16y
62. 14y3 94y2 28y
63.
4yz3 5yz2 6yz
64.
12a3 14a2 48a
65. 42a3 45a2 27a
66.
36p2 9p3 p4
67.
9x2y 30xy2 25y3
68. 8x2y 38xy2 35y3
69.
9x3y 24x2y2 16xy3
70.
9x3y 12x2y 4xy
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256
Section 5.3 / Factoring Polynomials of the Form ax 2 bx c
Objective B
257
To factor a trinomial of the form
ax 2 bx c by grouping
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For Exercises 71 to 130, factor by grouping.
71. 6x2 17x 12
72. 15x2 19x 6
73. 5b2 33b 14
74. 8x2 30x 25
75. 6a2 7a 24
76. 14a2 15a 9
77. 4z2 11z 6
78. 6z2 25z 14
79. 22p2 51p 10
80. 14p2 41p 15
81. 8y2 17y 9
82. 12y2 145y 12
83. 18t2 9t 5
84. 12t2 28t 5
85. 6b2 71b 12
86. 8b2 65b 8
87. 9x2 12x 4
88. 25x2 30x 9
89. 6b2 13b 6
90. 20b2 37b 15
91. 33b2 34b 35
92. 15b2 43b 22
93. 18y2 39y 20
94. 24y2 41y 12
95. 15a2 26a 21
96. 6a2 23a 21
97. 8y2 26y 15
98. 18y2 27y 4
99. 8z2 2z 15
100.
10z2 3z 4
101.
15x2 82x 24
102.
13z2 49z 8
103. 10z2 29z 10
104.
15z2 44z 32
105.
36z2 72z 35
106.
16z2 8z 35
107. 3x2 xy 2y2
108.
6x2 10xy 4y2
109.
3a2 5ab 2b2
110.
2a2 9ab 9b2
258
Chapter 5 / Factoring
111. 4y2 11yz 6z2
112.
2y2 7yz 5z2
113.
28 3z z2
114.
15 2z z2
115. 8 7x x2
116.
12 11x x2
117.
9x2 33x 60
118.
16x2 16x 12
119. 24x2 52x 24
120.
60x2 95x 20
121.
35a4 9a3 2a2
122. 15a4 26a3 7a2
123.
15b2 115b 70
124.
25b2 35b 30
125. 3x2 26xy 35y2
126.
4x2 16xy 15y2
127.
216y2 3y 3
128. 360y2 4y 4
129.
21 20x x2
130.
18 17x x2
APPLYING THE CONCEPTS
131.
In your own words, explain how the signs of the last terms of the
two binomial factors of a trinomial are determined.
132. x 12 x 1 6
133.
x 22 3x 2 2
134.
y 32 5y 3 6
135. 2y 22 y 2 3
136.
3a 22 a 2 4
137.
4y 12 7y 1 2
For Exercises 138 to 143, find all integers k such that the trinomial can be factored over the integers.
138. 2x2 kx 3
139.
2x2 kx 3
140.
3x2 kx 2
141. 3x2 kx 2
142.
2x2 kx 5
143.
2x2 kx 5
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For Exercises 132 to 137, factor.