C H A P T E R 5
Polynomials and Factoring
Section 5.1
Integer Exponents and Scientific Notation . . . . . . . . . 172
Section 5.2
Adding and Subtracting Polynomials
Section 5.3
Multiplying Polynomials . . . . . . . . . . . . . . . . . . 181
Section 5.4
Factoring by Grouping and Special Forms . . . . . . . . . 185
Section 5.5
Factoring Trinomials . . . . . . . . . . . . . . . . . . . . 189
Section 5.6
Solving Polynomial Equations by Factoring . . . . . . . . 192
Review Exercises
. . . . . . . . . . . 176
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
C H A P T E R 5
Polynomials and Factoring
Section 5.1
Integer Exponents and Scientific Notation
Solutions to Even-Numbered Exercises
2. (a) 52 y 4 y 2 25 y 4 y 2 25y 42 25y 6
(b) 5y2
4. (a) 5z32 52z32 25z 3 2 25z 6
y 4 25y 2 y 4 25 y 2 y 4 25y 24 25y 6
6. (a) 6xy7x 6 x x y 7 6x11y 7 6x 2 y 7
(b) x 5y32y3 2 x 5
(b) 5z4 54z 4 625z 4
8. (a) 3y32y2 33y32y2
y 33 2x 5y 6
27 2
y 3 y2
54y 32
54y 5
3
(b) 3y
2y 2 3 2 y 3 y 2
6y 32
6y 5
10. (a) m 3n2mn3 m3n2mn3 m 31 n23
12. (a)
28x2y3 28
2xy2
2
m 4n5
(b) m3n22 mn3 m32 n22 mn3
m 6n 41mn3
28
2
x2
x
y3
y2
x 21 y 32
14xy
m 6mn 4n3
(b)
m 61n 43
24xy2 24
8y
8
m 7n7
24
8
x
y2
y
x y 21
3xy
14. (a)
2a3y (b) 18. (a)
5
25a5
32a5
5 5 3 y
243y5
2a3y 23 ay
3x22x2
2 2
2 2
2x6x 2
16. (a)
4a2
9y2
3x24x2
12x2
12x
12x
2
4
2
(b)
2
(b)
4xy3 43x3y3 64x3y3
8x2y
8xy2
8xy2
8xy2
x 4y 4
x2y2
xy4
14x 4y 4
3xy2
3x2y2
3x 2y2
3
3x22x4
3x216x 4
2x 6x 4x 6x 2
2
2
48x
24x 3
x22
2x 32
x4
22 x32
4x 6
172
6 2
2
Section 5.1
20. (a)
x 3n y 2n1
x 3nny2n1 n3 x2n y2n1n3 x2ny n4
x n y n3
Integer Exponents and Scientific Notation
(b)
173
x 4n6y n10
x 4n6 2n5y n10 n2
x 2n5y n2
x 4n62n5y n10n2 x 2n1y12
22. 24 30. 1
1
24 16
24. 202 1
202
1
400
1
62 36
62
36. 42 43 42(3) 41 1
4
42. 53 543 53(4)3
3
26. 250 1
32.
45
38.
1
1
1
51
2(1) 3 52
5
5
125
54
3
28.
125
64
44. 412 42 16
2
1
82
64
82 1
34.
85
58
40.
105
105(6) 101 10
106
46. 4 32 4 513
4
53
125
48.
12 32
1
1233 3222
3 4
6 6
61
1
50. 32 430 1
52. x2
2
1
32
1
9
36 1 35
9
9
9
x5 x2(5)
x7
1
1
x7
1
61
6
54. t 1
t 6 t 1(6)
56. 3y3 3
y3
58. 5u2 1
5u2
1
25u 2
t 7
1
7
t
62.
6u2
15u1 3
23
5u1(2)
64.
5u4
5u40
5u0
60.
2
5u
68. 5s5t 56s2t 4 30s 5 2t 5 4 30s3t 1
30s3
t
4
4y
y1
66. 4a23 43a 6
5u4
2
1
5u
a6
a6
3
4
64
1
1
5u4 625u 4
70. 4y3z3 1
1
4y3z3
64y9z 3
64
25
y9
64z 3
174
72.
Chapter 5
4z 2
3
78.
Polynomials and Factoring
42
z2
z2
42
z2
16
74.
ab ba ba ba 3
3
3
3
3
3
b33
a33
b6
a6
x12y 8
24
x12y8
16
1 1
u1 v1
u
v
90. 1
u v1
1 1
u
v
1 1
u
v
1 1
u
v
5 x
1z33
2y2
x5 y 1z 0
2y2
5yx 1
2y2
5y 4
x2
80. ab2a2b21 a2b2a2b2
xy
5125xy
2 31y31 1
53
2 4 1
32
2
1
4
82. x53x0y 47y0 x53y 41
3x5y 4
a2(2)b2(2)
24x12 y8
2 3 3 1
76.
a4b4
84. 2x3y222 2x3y24
2y1z3
2z3
3
4yz
4yyz3
86.
1
a 4b4
5x2y51 51x2y5
2x5y 4
2x5y 4
88. x2x 2 y 2 x22 x2y 2
x2(5)y 54
52
x0 y2
x2
x 3y
10
1
y2
x2
92. 98,100,000 9.81 107
94. 956,300,000 9.563 108
uv
uv
vu
vu
96. 0.00625 6.25 103
98. 0.0007384 7.384 104
102. 0.0000001 1.0 107
104. 0.00003937 3.937 105
106. 5.05 1012 5,050,000,000,000
108. 8.6 109 0.0000000086
110. 3.5 108 350,000,000
112. 9.0 109 0.000000009
114. 9.0 104 0.0009
100. 139,500,000 1.395 108
116. 6.5 1062 104 6.52 1064
13.0 1010
1.3 1011
Section 5.1
118. 4 1063 43 106 3
120.
64 1018
Integer Exponents and Scientific Notation
2.5 103
0.5 1032
5 102
0.5 105
6.4 10
19
5.0 106
122. 62,000,0000.0002 6.2 1072 104
124.
72,000,000,000
7.2 1010
0.00012
1.2 104
12.4 107(4)
12.4 6.0 1010(4)
103
6.0 1014
1.24 104
126.
3,450,000,0000.000125 3.45 1091.25 104
52,000,0000.000003
5.2 1073 106
128.
14.5924 1010
3.82 1052
4
3
8.5 10 5.2 10 44.2 101
3.45 1.25 10 9(4)
5.2 3 107(6)
0.3301447964 109
3.301447964 108
105
4.3125 15.6 101
3.30 108
0.276 1051
0.276 10 4
2.76 103
130. 8.67 10 47 8.677 1028
132.
3682423.07 1028
6,200,0000.0053 6.2 106 5.0 1033
0.000355
3.5 1045
3.68 1034
6.2 106 5.03 109
3.55 1020
6.2125 103
3.55 1020
6.2125 103(20)
3.55
6.2(125 1017
3.55
775 1017
525.21875
1.48 1017
134. 8.483 10 22 84,830,000,000,000,000,000,000
136. 959.46 1015 899 1015
8.99 1017 meters
138. 752001.1 105 16,500 105 0.165 foot
140.
58 million
58,000,000
6 billion
6,000,000,000
5.8 10 7
6 10 9
0.966 102
9.66 103 tons
1
19 3 pounds
175
176
Chapter 5
142. 2x4 Polynomials and Factoring
1
1
Both the 2 and x are raised to the 4 power.
24 x 4 16x 4
2x4 2
x4
Only the x is raised to the 4 power.
144. 32.5 105 is not in scientific notation because 32.5 is not in the interval 1, 10.
Section 5.2
Adding and Subtracting Polynomials
4. Standard form: x 50
2. Standard form: t 5 14t 4 20t 4
Degree: 5
Degree: 1
Leading coefficient: 1
Leading coefficient: 1
8. Standard form: 4t 5 t 2 6t 3
6. Standard form: y 2 4y 12
Degree: 2
Degree: 5
Leading coefficient: 1
Leading coefficient: 4
12. Standard form: 12 at 2 48
10. Standard form: 28
Degree: 0
Degree: 2
Leading coefficient: 28
Leading coefficient: 12 a
14. 6y 3 y3 is a trinomial
18. 25 2u2 is a binomial
16. t 3 is a monomial
20. A trinomial of degree 4 and leading coefficient of 2 is
any trinomial beginning 2x 4 and containing two other
terms of degree less than 4 such as 2x 4 3x 2.
22. A monomial of degree 0 is any constant such as 16 or 8
or 4.
24. x3 4x13 is not a polynomial because the second term
is not of the form ax k (k must be a nonnegative integer).
26.
28. 6 2x 4x 6 2x 4x 2x 6
30. 3x 1 6x 1 3x 6x 11 9x
2
is not a polynomial because the expression is not
x4
of the form ax k.
32. 3x3 2x 8 3x 5 3x3 2x 3x 8 5 3x3 x 3
34. z3 6z 2 (3z2 6z z3 3z2 6z 6z 2
z3 3z2 2
36. y5 4y 3y y5 y5 5 y5 y5 y5 4y 3y 5
y5 y 5
38. 3a2 5a 7 a2 5a 2a2 8 3a2 a2 2a2 5a 5a 7 8
4a2 15
Section 5.2
Adding and Subtracting Polynomials
1
1
3
1
1
3
40. 2 4 y 2 y 4 3 y 4 2 y 2 3 y 4 3 y 4 4 y 2 2 y 2 2 3
33 y 4 13 y 4 14 y 2 64 y 2 1
43 y 4 74 y 2 1
42. 0.13x4 2.25x 1.63 5.3x4 1.76x2 1.29x 0.13x4 5.3x4 1.76x2 2.25x 1.29x 1.63 5.43x4 1.76x2 0.96x 1.63
44.
3x4 2x2 9
5x4 x2
x3 3x2
2x4 x2 9
50.
48. 16t2 48t 64
46. 4x3 8x2 5x 3
32t 16
7
16t2 16t 80
5x3 5x2 5x 4
1.7y3 6.2y2
52. 5y4 2 3y4 2 5y4 2 3y4 2
5.9
3.5y3 6.7y2 2.2y
5y4 3y4 2 2
1.8y3 0.5y2 2.2y 5.9
2y4 4
54. 5q2 3q 5 4q2 3q 10 5q2 3q 5 4q2 3q 10
5q2 4q2 3q 3q 5 10
q2 15
56. 10s2 5 2s2 6s 10s2 5 2s2 6s
10s2 2s2 6s 5
12s 2 6s 5
58.
12 32 x 21 x x
2
3
1
2
1
1
3x2 x 12 x x2 x3 3x2 x
6
3
2
6
12 23 x 16 x 12 x
12 46 x 16 x 12 x
2
2
3x2 x3
6
x2 x3
2
3
5
12 x x2 x3
6
2
1
5
12 x x2 x3
2
2
5
1
x 3 x 2 x 12
2
2
60. u3 9.75u2 0.12u 3 0.7u3 6.9u2 4.83 u3 9.75u2 0.12u 3 0.7u3 6.9u2 4.83
u3 0.7u3 9.75u2 6.9u2 0.12u 3 4.83
0.3u3 2.85u2 0.12u 1.83
177
178
Chapter 5
Polynomials and Factoring
62. y2 3y4 y4 y2 8y y2 3y4 y 4 y2 8y
y2 y2 3y4 y4 8y
2y4 2y2 8y
3t 4 5t 2 ⇒ 3t 4 5t 2
64.
4x2 5x 6 ⇒
66.
t 4 2t 2 14 ⇒ t 4 2t 2 14
4x2 5x 6
2x2 4x 5 ⇒ 2x2 4x 5
4t 4 7t 2 14
2x 2 9x 11
68. 13x3 9x2 4x 5 5x3 7x 3
70. 2x2 1 x2 2x 1 2x2 1 x2 2x 1
2x2 x2 2x 11
13x3 9x2 4x 5 ⇒ 13x3 9x2 4x 5
5x3
7x 3 ⇒ 5x3
x2 2x
7x 3
8x3 9x2 3x 8
72. 15 2y y2 3y2 6y 1 4y2 8y 16 y2 3y2 4y2 2y 6y 8y 15 1 16
0
74. p3 4 p2 4 3p 9 p3 4) p2 3p 4 9
p3 4) p2 3p 5
p3 4) p2 3p 5
p3 p2 3p 4 5
p3 p2 3p 9
76. 5x4 3x2 9 2x4 x3 7x2 x2 6 5x4 3x2 9 2x4 x3 7x2 x2 6
5x4 2x4 x3 3x2 7x2 x2 9 6
3x4 x3 5x2 15
78. x3 2x2 x 52x3 x2 4x x3 2x2 x 10x3 5x2 20x
x3 10x3 2x2 5x2 x 20x
9x3 7x2 19x
80. 10v 2 8v 1 3v 9 10v 20 8v 8 3v 27
10v 8v 3v 20 8 27
5v 1
82. 97x2 3x 3 415x 2 3x2 7x 63x2 27x 27 60x 8 3x2 7x
63x2 3x2 27x 60x 7x 27 8
60x2 80x 19
84. 3x2 23x 9 x2 3x2 23x 9 x2
3x2 6x 18 2x2
3x2 2x2 6x 18
5x2 6x 18
Section 5.2
Adding and Subtracting Polynomials
86. 6x 2r 5x r 4 2x 2r 2x r 3 6x 2r 2x 2r 5x r 2x r 4 3
8x 2r 3x r 7
88. x 2m 6x m 4 2x 2m 4x m 3 x 2m 6x m 4 2x 2m 4x m 3
x 2m 2x 2m 6x m 4x m 4 3
x 2m 2x m 7
90. 4x 3n 5x 2n x n x 2n 9x n 14 4x 3n 5x 2n x n x 2n 9x n 14
4x 3n 5x 2n x 2n x n 9x n 14
y2
X,T, 3
1
2
1
2
X,T, 3 2 X,T, >
Y
X,T, 3
X,T, x2
ENTER
X,T, 3
>
92. y1
>
4x 3n 4x 2n 8x n 14
X,T, 5
x2
X,T, 1 GRAPH
y1 and y2 represent equivalent
expressions since the graphs of
y1 and y2 are identical.
−6
6
−3
94. hx f x gx
4x3 3x2 7 9 x x2 5x3
4x3 3x2 7 9 x x2 5x3
4x3 5x3 3x2 x2 x 7 9
9x3 2x2 x 2
96. Polynomial
ht 16t 2 256
Value
Substitute
Simplify
at 0
1602 256
256 feet
bt 1
161 256
240 feet
ct 16
256
156 feet
1642 256
0 feet
5
2
dt 4
2
5 2
2
At time t 0, the object is dropped from a height of 256 feet and continues to fall, reaching the ground at time t 4.
98. Polynomial
ht 16t 2 96t
Value
Substitute
at 0
1602 960
0 feet
bt 2
162 962
128 feet
ct 3
163 963
144 feet
dt 6
1662 966
0 feet
2
2
Simplify
At t 0, the object is projected upward from a height of 0 feet (on the ground), reaches a maximum height, and returns
downward. At time t 6, the object is again on the ground.
179
180
Chapter 5
Polynomials and Factoring
102. The free-falling object was thrown upward.
100. The free-falling object was thrown upward.
ht 160 500
h0 1602 320 300
h0 0 0
h0 0 0 300
h0 0 feet
h0 300 feet
2
104. ht 16t2 200
h1 1612 200 16 200 184 feet
h2 1622 200 164 200 64 200 136 feet
h3 1632 200 169 200 144 200 56 feet
Profit
106. Verbal Model:
Revenue Cost
PRC
Equation:
P 17x 12x 8000
P 5x 8000
P 510,000 8000
P $42,000
108. Perimeter 4x 2 2x 10 2x 5 x 3 2x 4x
15x 10
110. Area of Region 5 x 5 x 5 3x or 5 x x 3x
5x 5x 15x
or 55x
25x
or 25x
112. Area 4x 75 2x4
20x 35 8x
12x 35
114. (a) Verbal Model:
Equation:
Per capita
consumption of
milk and coffee
Per capita
consumption
of milk
Per capita
consumption
of coffee
y 0.33t 25.9 0.043t 2 0.33t 18.5
y 0.043t 2 0.33t 0.33t 25.9 18.5
y 0.043t 2 44.4
(b) Keystrokes:
Y
0.043 X,T, x2
44.4 GRAPH
50
6
10
45
Consumption was increasing over the interval 6 ≤ t ≤ 10.
Section 5.3
Multiplying Polynomials
181
116. Addition (or subtraction) separates terms.
Multiplication separates factors.
118. Yes, two third-degree polynomials can be added to produce a second-degree polynomial. For example,
x 3 2x 2 4 x 3 x 2 3x x 2 3x 4
120. To subtract one polynomial from another, add the opposite. You can do this by changing the sign of each of the terms of the
polynomial that is being subtracted and the adding the resulting like terms.
Section 5.3
Multiplying Polynomials
2. 6n3n2 6 3n
n2 18n12 18n3
6. 3y3y2 7y 3 3y3y2 3y7y 3y3
4. 5z2z 7 5z2z 5z7 10z2 35z
8. 3a28 2a a2 3a28 3a22a 3a2a2
9y3 21y2 9y
10. y47y3 4y2 y 4 7y7 4y6 y5 4y4
3a 4 6a 3 24a2
12. 4t3tt2 1 12t2t2 1
12t2t2 12t21
12t4 12t2
14. ab32a 9a2b 3b ab32a ab39a2b ab33b
16. x 5x 3 x2 3x 5x 15 x2 8x 15
2a2b3 9a3b4 3ab4
18. x 7x 1 x2 x 7x 7 x2 6x 7
20. x 6x 6 x2 6x 6x 36 x2 36
22. 3x 1x 4 3x2 12x x 4 3x2 11x 4
24. 4x 73x 7 12x2 28x 21x 49 12x2 49x 49
26. 6x2 29 2x 54x2 12x3 18 4x 12x3 54x2 4x 18
3
3
28. 5t 4 2t 16 10t2 80t 2t 12
30. 2x y3x 2y 6x2 4xy 3xy 2y2
3
10t2 160
2 t 2 t 12
163
10t2 2 t 12
32. s 3ts t s 3ts t s2 st 3ts 3t2 s2 st 3ts 3t2
2st 6t2
34. z 2z2 4z 4 z 2z2 z 24z z 24
z3 2z2 4z2 8z 4z 8
z3 2z2 4z 8
6x2 7xy 2y2
182
Chapter 5
Polynomials and Factoring
36. 2t 3t2 5t 1 2t 3t2 2t 35t 2t 31
2t3 3t2 10t2 15t 2t 3
2t3 7t2 13t 3
38. 2x2 5x 13x 4 3x 42x2 3x 45x 3x 41
6x3 8x2 15x2 20x 3x 4
6x3 23x2 23x 4
40. x2 4x2 2x 4 x2 4x2 x2 42x x2 44
x2
x2 4x2 x22x 8x 4x2 16
x4 4x2 2x3 8x 4x2 16
x4 2x3 8x 16
42. 2x2 32x2 2x 3 2x2 32x2 2x2 32x 2x2 33
4x4 6x2 4x3 6x 6x2 9
4x4 4x3 6x 9
44. y2 3y 52y2 3y 1 y22y2 3y 1 3y2y2 3y 1 52y2 3y 1
2y4 3y3 y2 6y3 9y2 3y 10y2 15y 5
2y4 3y3 18y 5
46.
50.
4x4 6x2
2x2
4
12x 18x2
6 12x4 18x2
8x
8x6
9
3
27
48.
z2 z 1
z2
2z2 2z 2
z3 z2 z
z3 z2 z 2
52.
y2
2y2
y2
3
3y 9y2
4
2y 6y3 10y2
2y4 3y3
27
2s2 5s 6
3s 4
8s2 20s 24
6s3 15s2 18s
6s3 23s2 38s 24
3y 5
3y 1
3y 5
15y
18y 5
54. x 5x 5 x2 52 x2 25
56. x 1x 1 x2 12 x2 1
58. 4 3z4 3z 42 3z2 16 9z2
60. 8 3x8 3x 82 3x2 64 9x2
62. 5u 12v5u 12v 5u2 12v2
64. 8x 5y8x 5y 8x2 5y2
25u2 144v2
66.
23x 723x 7 23x2 72 49 x 2 49
64x2 25y2
68. 4a 0.1b4a 0.1b 4a2 0.1b2
16a2 0.01b 2
Section 5.3
Multiplying Polynomials
183
70. x 22 x2 2x2 22 x2 4x 4
72. u 72 u2 2u7 72 u2 14u 49
74. 3x 82 3x2 23x8 82 9x2 48x 64
76. 5 3z2 52 253z 3z2 25 30z 9z2
78. 3m 4n2 3m2 23m4n 4n2
80. x 4 y2 x 42 2x 4(y y2
x2 2x4 42 2xy 8y y2
9m 2 24mn 16 n 2
x2 8x 16 2xy 8y y2 or
x 2 2xy y 2 8x 8y 16
82. z y 1z y 1 z2 y 12
84. y 23 y 2 y 2 y 2
z2 y2 2 y1 12
y2 4y 4 y 2
z2 y2 2y 1
y2 y 2 4y y 2 4 y 2
z2 y2 2y 1
y3 2y2 4y2 8y 4y 8
y3 6y2 12y 8
86. u v3 u vu vu v
88. 5x r 4x r2 3x r 5x r 4x r2 5x r 3x r u2 2uv v2u v
20x rr2 15x rr
u2u v 2uvu v v2u v
20x 2r2 15x2r
u3 u2v 2u2v 2uv2 uv2 v3
u3 3u2v 3uv2 v3
90. x3m x2mx2m 2x 4m x3mx2m 2x 4m x2m x2m 2x 4m
x3m2m 2x3m4m x2m2m 2x2m4m
x 5m 2x7m x 4m 2x 6m
2x7m 2x 6m x 5m x 4m
92. y mnmn y mnmn
mn n2
ym
2 2
ym
2
2mnn2
94. Keystrokes:
y1
Y
y2
X,T, 6
X,T,
x2
3
ENTER
x2
6 X,T, 9 GRAPH
−4
y1 y2 because x 32 (x2 2x3 32 x2 6x 9
8
−2
96. Keystrokes:
y1
Y
y2
X,T, 6
X,T, x2
y1 y2 because x 1
2
1
1
4
2
X,T, 1
2
ENTER
GRAPH
x (x
1
2
2
12 x2 14
2
−6
6
−2
184
Chapter 5
Polynomials and Factoring
98. (a) f y 2 2 y 22 5 y 2 4
2 y2 4y 4 5y 10 4
2y2 8y 8 5y 10 4
2y2 3y 2
(b) f 1 h f 1 21 h2 51 h 4 212 51 4
21 2h h2 5 5h 4 2 5 4
2 4h 2h2 5 5h 4 1
2h2 h
100. (a) Verbal Model: Volume Length
Width
Height
Function: Vn 2n 2 2n 2 2n
4n2 42n
8n3 8n
(b) 2n 2 6, n 4, V(4) 843 84
864 32
512 32
480 cubic inches
(c) Verbal Model:
Area
Length
Width
Function: An 2n 2 2n 2
4n2 4
Area 2n 2 22n 2 2
(d)
2n 42n
4n2 8n
An 4 2n 4 2 2n 4 2
2n 8 22n 8 2
2n 102n 6
4n2 12n 20n 60
4n2 32n 60
Area of
Area of
Area of
Inside
102. Verbal Model: Shaded Outside Region
Rectangle
Rectangle
Alwlw
Equation:
A 3x x x 32
A 3x2 2x 6
104. Verbal Model:
Equation:
Area of
Area of
Area of
shaded region
triangle
rectangle
A x 5x 2x 12xx 5
x 53x) 12xx 5
3x2 15x 12x2 52x
5
25
2x2 2 x
1
106. A 2b
h
1
2 (3xx
5
32xx 5
32x2 15
2x
Section 5.4
108. Verbal Model: Revenue
Number of
units sold
Price per
unit
Factoring by Grouping and Special Forms
110. Interest 10001 0.0952
10001.0952
Rxp
Equation:
185
10001.199025
x 20 0.015x
1199.025
20x 0.015x 2
$1199.03
R 0.015x 2 20x
R 0.015502 2050
$962.50
112. Area l w
114. (a) x y2 x yx y
x ax a
x2 xy yx y2
x2 2ax a2
x2 2xy y2
Area x x a
x a x a a
(b) x y2 x yx y
x2 ax ax a2
x2 xy xy y2
x2 2ax a2
x2 2xy y2
Formula x ax a x2 2ax a2
(c) x yx y x2 xy xy y2
x2 y2
Square of a binomial
116. Example: x 2x 3 xx 3 2x 3
x2 3x 2x 6
118. The degree of the product of two polynomials of degrees
m and n is m n.
x2 x 6
Section 5.4
2.
36 22
Factoring by Grouping and Special Forms
32
150 2 3
6. 45y 1 32
4. 27x4 33x4
18x3 2 32x3
52
100 22 52
150y3 2 3
GCF 9x3
GCF 3
5y
5 2 y3
5 y 15y
GCF 2
10. 663 y 2 3
8. 16x2y 24x2y
22
37x
36x2y2 22
32 x2 y2
84xy2
y2
11 3 y
11 3 y2
443 y 22
GCF 2 11
3 y 223 y
2
GCF 22 x y 4xy
12. 7y 7 7 y 1
14. 9x 30 33x 10
16. 54x2 36 183x2 2
18. y2 5y y y 5
20. 3x2 6x 3x x 2
22. 16 3y3 is prime.
No common factor other than 1.
24. 9 27y 15y2 33 9y 5y2
26. 4uv 6u2v2 2uv2 3uv
28. 4x2 2xy 3y2 Prime
No common factor other than 1.
186
Chapter 5
Polynomials and Factoring
30. 17x5y3 xy2 34y2 y217x5y x 34
32. 4 x3 4 x3 x3 4
34. 15 5x 53 x 5x 3
36. 12x 6x2 18 6 2x x2 3
6 x 2 2x 3
38. 2t3 4t2 7 12t3 4t2 7
40. 3z 38 1824z 3
2t3 4t2 7
42. 13x 56 162x 5
44. 7ts 9 6s 9 s 97t 6
46. 64t 3 5t4t 3 4t 36 5t
48. 45y 12 3y25y 12 5y 124 3y2
50. 2y2 y2 63 7 y2 63 y2 632y2 7
52. 3x 72x 1 x 62x 1 2x 13x 7 x 6
2x 14x 1
54. x2 9x x 9 x2 9x x 9
56. y2 3y 4y 12 y2 3y 4y 12
xx 9 x 9
y y 3 4 y 3
x 9x 1
y 3 y 4
58. t3 11t2 t 11 t3 11t2 t 11
60. 3s3 6s2 5s 10 3s3 6s2 5s 10
t2t 11 t 11
3s2s 2 5s 2
t 11t2 1
s 23s2 5
62. 4u4 6u 2u3 3 4u4 6u 2u3 3
2u2u3 3 12u3 3
2u3 32u 1
64. 10u4 8u2v3 12v4 15u2v 10u4 8u2v3 12v4 15u2v
2u25u2 4v3 3v4v3 5u2
5u2 4v32u2 3v
66. y2 144 y2 122
y 12 y 12
68. 16 b2 42 b2
4 b4 b
70. 9z2 36 3z2 62
3z 63z 6
3z 23z 2
9z 2z 2
72. 49 64x2 72 8x2
7 8x7 8x
74. 9u2 v2 3u2 v2
3u v3u v
76. 100a2 49b2 10a2 7b2
10a 7b10a 7b
Section 5.4
Factoring by Grouping and Special Forms
187
1
36
1
6
80. 4x2 49y2 2x 7y
9
78. v2 25
v2 35 2
2
2x 7y2x 7y
v 5 v 5 3
2
1
3
6
1
6
84. 36 y 62 62 y 62
82. x 32 4 x 32 22
x 3 2x 3 2
6 y 66 y 6
x 5x 1
6 y 66 y 6
y 12 y y12 y
86. 3y 12 x 62 3y 1 x 63y 1 x 6
3y 1 x 63y 1 x 6
3y x 73y x 5
or
x 3y 7x 3y 5
88. t 3 1 t 3 13
90. z3 125 z3 53
t 1t 2 t 1
92. 27s3 64 3s3 43
3s 49s2 12s 16
96. m3 8n3 m3 2n3
z 5z2 5z 25
94. 64v3 125 4v3 53
4v 516v2 20v 25
98. u3 125v3 u3 5v3
m 2nm2 2mn 4n2
100. 8y2 18 24y2 9
u 5vu2 5uv 25v2
102. a3 16a aa2 16
22y2 32
aa2 42
22y 32y 3
aa 4a 4
104. u4 16 u22 42
106. 6x5 30x3 6x3x2 5
u2 4u2 4
u2 22u2 4
u 2u 2u2 4
108. 2u6 54v6 2u6 27v6
110. 81 16y4n 92 4y2n2
2u23 3v23
9 4y2n9 4y2n
2u2 3v2u4 3u2v2 9v4
32 2yn29 4y2n
3 2yn3 2yn9 4y2n
112. 3x n1 6x n 15x n2 3x n x 2 5x 2
3x n 5x 2 x 2
114. x 2rs 5x r3s 10x 2r2s x rsx r 5x 2s 10x rs
x rs10x rs 5x 2s x r
188
Chapter 5
Polynomials and Factoring
116. Keystrokes:
4
Y
y2
X,T, x2
2 X,T, >
X,T, y1
3
X,T, 2
x2
ENTER
−6
GRAPH
6
y1 y2
−4
118. Keystrokes:
y1
y2
X,T, Y
4
X,T, X,T, 1
1
X,T, 4
X,T, 4
1
ENTER
−8
10
GRAPH
y1 y2
−8
120. 6x3 8x2 9x 12 6x3 8x2 9x 12
or 6x3 8x2 9x 12 6x3 9x 8x2 12
2x23x 4 33x 4
3x2x2 3 42x2 3
3x 42x2 3
2x2 33x 4
124. kQx kx2 kxQ x
122. R 1000x 0.4x2
126. A 32w w2
x1000 0.4x
w32 w
rr 2h
wl
R xp
p 1000 0.4x
130. (a) Solid
128. S r2 2rh
Thus, l 32 w
Length
Width
Height
Volume
Entire cube
a
a
a
a3
Solid I
a
a
ab
a2a b
Solid II
a
ab
b
aba b
Solid III
ab
b
b
b2a b
Solid IV
b
b
b
b3
(b) Solid I Solid II Solid III
a2a b aba b b2a b a ba2 ab b2
(c) If the smaller cube is removed from the larger, the remaining solid has a volume of a3 b3 and is composed of the three
rectangular boxes labeled solid I, solid II, and solid III. From part (b) we have a3 b3 a ba2 ab b2.
132. Check a result after factoring by multiplying the factors to see if the product is the original polynomial.
134. Factor used as a noun expresses any one of the expressions that when multiplied together yields the product.
Factor used as a verb is the process of finding the expressions that, when multiplied together, yield the given product.
136. An example of a polynomial that is prime with respect to the integers is x2 1.
Section 5.5
Section 5.5
Factoring Trinomials
Factoring Trinomials
2. z2 6z 9 z2 23z 32
4. y2 14y 49 y2 27y 72
z 3z 3
y 72
z 32
8. 4x2 4x 1 2x2 22x1 12
6. 4z2 28z 49 2z2 22z7 72
2x 12
2z 72z 7
2z 72
10. x2 14xy 49y2 x2 2x7y 7y2
12. 4y2 20yz 25z2 2y2 22y5z 5z2
x 7y2
2y 5z2
14. 4x2 32x 64 4x2 8x 16
16. 3u3 48u2 192u 3uu2 16u 64
4x2 2x4 42
3uu2 2u8 82
4x 42
3uu 82
18. 18y3 12y2 2y 2y9y2 6y 1
2y3y2 23y1 12
2y3y 12
1
8
16
1
1
4
4
1
4
20. 9x2 15x 25 3x 23x5 5 3x 5 2
2
2
or
25 2
144
225
x 120
225 x 225
1
225
25x2 120x 144
1
1
225
5x2 25x12 122 225
5x 122
9
22. x2 bx 16
x2 bx 34 24. 16x2 bxy 25y2
2
b 32
(a)
x2 32
b 40
(a)
x 169 x2 234 x 34 16x2 40xy 25y2 4x2 24x5y 5y2
2
4x 5y2
x2 32x 169
or
or
b 32
(b)
b 40
(b)
9
x2 32x 16
x2 2 34 x 34 2
16x2 40xy 25y2 4x2 24x5y 5y2
4x 5y2
3
9
x2 2x 16
26. x2 12x c
28. z2 20z c
c 36
x2
c 100
12x 36 (x)2
26x 6
x 62
2
z2 20z 100 (z)2 210z 102
z 102
189
190
Chapter 5
Polynomials and Factoring
30. a2 2a 8 a 4a 2
32. y2 6y 8 y 4 y 2
34. x2 7x 12 x 4x 3
36. z2 2z 24 z 4z 6
38. x2 7x 10 x 5x 2
40. x2 10x 24 x 6x 4
42. m2 3m 10 m 5m 2
44. x2 4x 12 x 6x 2
46. y2 35y 300 y 15 y 20
48. u2 5uv 6v2 u 3vu 2v
50. a2 21ab 110b2 a 10ba 11b
52. b 15
54. b 6
b 6
58. c 8
x2 6x 7 x 7x 1
x2 6x 7 x 7x 1
x2 9x 8 x 8x 1
x2 15x 14 x 14x 1
b 15
x2 15x 14 x 14x 1
x9
x2 9x 14 x 7x 2
x 9
x2 9x 14 x 7x 2
56. b 17
x2 17x 38 x 19x 2
b 17
x2 17x 38 x 19x 2
b 37
x2 37x 38 x 38x 1
b 37
x2 37x 38 x 38x 1
60. There are many possibilities such as:
c 14
x2 9x 14 x 7x 2
c 11
x2 12x 11 x 11x 1
c 10
x2 9x 10 x 10x 1
c 20
x2 12x 20 x 2x 10
c 36
x2 9x 36 x 12x 3
c 27
x2 12x 27 x 9x 3
There are more possibilities.
62. 5x2 19x 12 x 35x 4
Also note that if c is a negative number, there are many
possibilities for c such as the following:
c 13
x2 12x 13 x 13x 1
c 28
x2 12x 28 x 14x 2
c 45
x2 12x 45 x 15x 3
64. 5c2 11c 12 c 35c 4
66. 3y2 y 30 y 33y 10
68. 3x2 16x 12 3x 2x 6
70. 6x2 x 15 3x 52x 3
72. 3y2 10y 8 3y 4 y 2
74. 15x2 4x 3 5x 33x 1
76. 3z2 z 4 3z 4z 1
78. 10x2 24x 18 25x2 12x 9
80. 20x2 x 12 5x 44x 3
25x 3x 3
82. 6x2 5x 6 is Prime
84. 2 5x 12x2 2 3x1 4x
86. 12x2 32x 12 43x2 8x 3
88. 12x2 42x3 54x4 6x22 7x 9x2
43x 1x 3
6x22 9x1 x
Section 5.5
Factoring Trinomials
90. 6u2 5uv 4v2 3u 4v2u v
92. 10x2 9xy 9y2 5x 3y2x 3y
94. 2x2 9x 9 2x2 6x 3x 9
96. 6x2 x 15 6x2 10x 9x 15
2xx 3 3x 3
2x3x 5 33x 5
x 32x 3
3x 52x 3
98. 12x2 28x 15 12x2 18x 10x 15
100. 20y2 45 54y2 9
6x2x 3 52x 3
52y2 32
6x 52x 3
52y 32y 3
102. 16z3 56z2 49z z16z2 56z 49
104. 3t3 24 3t3 8
z4z2 24z(7) 72
3t3 23
z4z 72
3t 2t2 2t 4
106. 8m3n 20m2n2 48mn3 4mn2m2 5mn 12n2
108. x3 7x2 4x 28 x3 7x2 4x 28
4mn2m 3nm 4n
x2x 7 4x 7
x 7x2 4
x 7x 2x 2
110. x 7y2 4a2 x 7y 2ax 7y 2a
112. a2 2ab b2 16 a2 2ab b2 16
x 7y 2ax 7y 2a
a b2 42
a b 4a b 4
116. y 2n y n 2 y n 2 y n 1
114. x4 16y4 x22 4y22
x2 4y2x2 4y2
x 2yx 2yx2 4y2
118. x 2n 4x n 12 x n 6x n 2
120. 3x 2n 16x n 12 3x n 2x n 6
122. Keystrokes:
y1
y2
Y
6
4 X,T, 2 X,T, x2
1
4 X,T, x2
1 ENTER
GRAPH
−4
y1 y2
8
−2
124. Keystrokes:
y1
y2
Y
3 X,T, 3 X,T, 4
x2
4
8 X,T, X,T, 16 ENTER
4
−18
24
GRAPH
y1 y2
−24
191
192
Chapter 5
Polynomials and Factoring
126. a2 2a 1 a 12 matches graph (a).
Area of
Shaded
Region
130. Verbal Model:
Area 12
Equation:
128. ab a b 1 a 1b 1 matches graph (d).
132. (a) 8n3 12n2 2n 3 8n3 12n2 2n 3
Area of
Area of
Larger Smaller
Triangle
Triangle
4n22n 3 12n 3
4n2 12n 3
54x 3x 3 12 5 4
58x 32 58
2n 12n 12n 3
16
(b) If n 15,
58x2 6x 9 16
2n 1 215 1 29
58x2 6x 7
2n 1 215 1 31
58x 7x 1
2n 3 215 3 33
134. An example of a prime trinomial is x2 x 1.
136. 3x 6 3x 2 and 3x 9 3x 3
so 3x 63x 9 3x 2 3x 3
9x 2x 3 not 3x 2x 3
138. 2x 4x 1 is not in completely factored form because 2x 4 has the common factor 2, 2x 4 2x 2
Section 5.6
Solving Polynomial Equations by Factoring
2. zz 6 0
z0
4. s 16s 15 0
z60
s 16 0
z 6
s 15 0
s 16
t30
s 15
t80
t3
t 8
10. 15xx 23x 4 0
8. 5x 3x 8 0
5x 3 0
x80
5x 3
x8
x
6. 17t 3t 8 0
1
5x
0
x20
x0
x2
3x 4 0
3x 4
3
5
x 43
12. y 392y 7 y 12 0
y 39 0
2y 7 0
y 39
2y 7
y
16. 4x2 6x 0
14. 3x2 9x 0
y 12 0
3xx 3 0
y 12
2x 0
2x 3 0
x0
2x 3
x
3
2
x30
x 3
8x 2 5x
18.
2x2x 3 0
x0
27
3x 2 7x
20.
8x 2 5x 0
3x 2 7x 0
x 8x 5 0
x 3x 7 0
x0
x0
8x 5 0
3x 7 0
8x 5
3x 7
x 58
x 73
Section 5.6
22. x2 121 0
Solving Polynomial Equations by Factoring
24. 25z2 100 0
26. x2 x 12 0
x 11x 11 0
25z2 4 0
x 4x 3 0
x 11 0
25z 2z 2 0
x40
x 11 0
x 11
x 11
z20
z20
z2
28. 20 9x x2 0
x30
x4
x 3
z 2
32. 11 32y 3y2 0
30. 14x2 9x 1
5 x4 x 0
14x2 9x 1 0
11 y1 3y 0
5x0
4x0
7x 12x 1 0
11 y 0
5x
4x
7x 1 0
x
2x 1 0
1
7
2x 15 x 2
34.
x
y 11
1
36. x 2 15 2x
x 5x 3 0
x 5x 3 0
x50
x30
x 5
x 3
38. a2 4a 10 6
3y 1
y 3
x 2 2x 15 0
x5
1 3y 0
21
x 2 2x 15 0
x50
193
40. x2 12x 21 15
x30
x3
42. 16t2 48t 40 4
a2 4a 4 0
x2 12x 36 0
16t2 48t 36 0
a 2a 2 0
x 6x 6 0
44t2 12t 9 0
a20
x60
42t 32t 3 0
a 2
2t 3 0
x6
2t 3
3
t 2
44. x x 15 3x 15 0
46. x x 10 2x 10 0
x 15x 3 0
x 10x 2 0
x 15 0
x 15
x 10
x 3
48. ss 4 96
s2
x 10 0
x30
50. xx 4 12
4s 96
x2
x20
x2
52. 3u3u 1 20
4x 12
9u2 3u 20
s2 4s 96 0
x2 4x 12 0
9u2 3u 20 0
s 12s 8 0
x 6x 2 0
3u 43u 5 0
s 12 0
s80
x60
3u 4 0
s8
x6
s 12
x20
x 2
3u 5 0
3u 4
3u 5
4
3
u 35
u
194
Chapter 5
Polynomials and Factoring
54. x 8x 7 20
56. u 6u 4 21
58. s 42 49 0
x2 7x 8x 56 20
u2 2u 24 21
s 4 7s 4 7 0
x2 15x 36 0
u2 2u 3 0
s 3s 11 0
x 12x 3 0
u 3u 1 0
s30
x 12 0
x30
u30
x3
u3
x 12
60. 1 y 32
u10
s 11 0
s3
s 11
u 1
62. 1 x 1 2 0
64. s 52 49 0
1 y 32 0
1 x 11 x 1 0
s 5 7s 5 7 0
1 y 31 y 3 0
xx 2 0
s 2s 12 0
y 2 y 4 0
x 0
s20
y 2 0
y40
y 2
x0
x20
x 2
s 12 0
s2
s 12
y 4
y 2
68. 3u 3 5u 2 2u
66. x3 18x2 45x 0
xx2 18x 45 0
3u3 5u2 2u 0
xx 15x 3 0
u3u2 5u 2 0
x0
x 15 0
x30
x 15
x 3
u3u 1u 2 0
u0
3u 1 0
u20
3u 1
u
70. 163 u u23 u 0
72. x3 2x2 4x 8 0
3 u16 u2 0
x2x 2 4x 2 0
3 u4 u4 u 0
x 2x2 4 0
3u0
4u0
3u
4u
4u0
u 4
74. v3 4v2 4v 16 0
x 2x 2x 2 0
x20
x20
x2
x2
x3x 2 9xx 2 0
v 4v2 4 0
x 2x3 9x 0
v 4v 2v 2 0
x 2xx2 9 0
v 4
v20
v2
x20
x 2
76. x4 2x3 9x2 18x 0
v2v 4 4v 4 0
v40
u2
1
3
v20
v 2
x 2xx 3x 3 0
x20
x 2
x0
x30
x3
x30
x 3
Section 5.6
78. 9x4 15x3 9x2 15x 0
Solving Polynomial Equations by Factoring
80. From the graph, the x-intercept is 2, 0.
The solution of the equation 0 x2 4x 4 is 2.
3x 5 3x3x 5 0
3x3
0 x2 4x 4
3x 53x3 3x 0
0 x 2x 2
3x 53xx2 1 0
3x 53xx 1x 1 0
3x 5 0
3x 0
x10
x 53
x0
x1
x20
x20
x2
x2
x10
x 1
82. From the graph, the x-intercepts are 1, 0, 1, 0, and
3, 0. The solutions to the equation are 1, 1, and 3.
84. Keystrokes:
Y
0 x3 3x2 x 3
X,T, x2
11 X,T, 28 GRAPH
The x-intercepts are 4 and 7, so the solutions are 4 and 7.
0 x2x 31x 3
10
0 x 3x 1
2
0 x 3x 1x 1
x30
x10
x3
x1
x10
0
x 1
10
−4
86. Keystrokes:
Y
4
X,T, 2
x2
9 GRAPH
−2
6
The x-intercepts are 1 and 5, so the solutions are 1 and 5.
−10
88. Keystrokes:
X,T, 8
>
Y
3
4 X,T, GRAPH
−4
The x-intercepts are 2 , 0, and 2, so the solutions are 2, 0, and 2.
4
−8
90. Keystrokes:
2 8
X,T, 2 X,T, x2
X,T, >
Y
3 GRAPH
The x-intercepts are 2, 1, and 1, so the solutions are 2, 1, and 1.
−4
4
−8
94. x 1 and x 6
92. ax2 ax 0
axx 1 0
x 1x 6 0
ax 0
x10
x2 7x 6 0
0
a
x1
x0
x1
x
195
196
Chapter 5
96. Verbal Model:
Polynomials and Factoring
First
Integer
Second
Integer
132
First integer x
Labels:
Second integer x 1
x x 1 132
Equation:
x 2 x 132 0
x 12x 11 0
x 12 0
x 11 0
x 12
x 11 1st integer
x 1 12 2nd integer
reject
98. Verbal Model:
Area of
Length
exposed picture
Width
100. Verbal Model: Area 12
Labels:
Length 28 2w
Labels:
Equation:
468 28 2w20 2w
Base x
70 12xx 4
0 12x2 2x 70
0 92 96w 4w2
0 x2 4x 140
0 423 24w w2
0 x 14x 10
0 21 w23 w
x 14 0
23 w 0
x 10 0
x 14
23 w
1w
Height
70 12x2 2x
468 560 56w 40w 4w2
1w0
Height x 4
Width 20 2w
Equation:
Base
x 10
reject
reject
The base of the triangle is 14 inches.
The width of the frame is 1 cm.
The height is 14 4 10 inches.
102. S x2 4xh
104.
880 x2 4x6
16t 2 576 0
16t 2 36 0
0 x2 24x 880
16t 6t 6 0
0 x 20x 44
t60
0 x 20
20 x
20" 20"
0 x 44
44 x
reject
t6
t60
t 6
Thus, the object reaches the ground after 6 seconds.
Section 5.6
106. h 16t 2 16t 32
Solving Polynomial Equations by Factoring
108.
0 16t 2 16t 32
h 16t 2 80t
96 16t 2 80t
0 16t 2 t 2
0 16t 2 80t 96
0 16t 2t 1
0 16t 2 5t 6
t20
0 16t 2t 3
t10
t2
t 1
t20
t30
t 2 seconds
reject
The object reaches the ground after 2 seconds.
t 3 seconds
The object reaches your friend on the way up at
2 seconds and at 3 seconds on the way down.
110. Verbal Model: Revenue Cost
60x x2 75 40x
Equation:
0 x2 20x 75
0 x 5x 15
x50
x 15 0
x 5 units
x 15 units
112. (a) 3x 62 10x 6 8 0
let u x 6
3u2
(b) 8x 22 18x 2 9 0
let u x 2
10u 8 0
8u2 18u 9 0
3u 2u 4 0
4u 32u 3 0
3u 2 0
u40
4u 3 0
23
u4
3
4
u 32
x 6 23
x64
x 2 34
x 2 32
u
x 20
3
x 2
u
2u 3 0
x 54
x 12
or
or
3x2 12x 36 10x 60 8 0
8x2 4x 4 18x 36 9 0
3x2 36x 108 10x 68 0
8x2 32x 32 18x 36 9 0
3x2 26x 40 0
8x2 14x 5 0
3x 20x 2 0
4x 52x 1 0
3x 20 0
4x 5 0
3x 20
x
20
3
x20
x 2
2x 1 0
4x 5
2x 1
5
4
x 12
x
197
198
Chapter 5
Polynomials and Factoring
114. An example of how the Zero-Factor Property can be
used to solve a quadratic equation: If xx 2 0, then
x 0 or x 2 0. The solutions are x 0 and x 2.
116. Yes, it is possible for a quadratic equation to have only
one solution. For example:
x2 2x 1 0
x 12 0
x 1
118. (a)
4
−9
9
−8
From the graph, the solution is between x 1 and x 2.
(b)
x
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
x3 x 3
3.0
2.769
2.472
2.103
1.656
1.125
0.504
0.213
1.032
1.959
3.0
From this table, the solution is between x 1.6 and x 1.7.
x
x x3
3
1.60
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.70
0.504
0.437
0.368
0.299
0.229
0.158
0.086
0.0125
0.062
0.137
0.213
An estimate of the solution from the table is x 1.67.
Review Exercises for Chapter 5
2. 3y2 y4 3y24 3y6
4. v42 v4 2 v8
6. 3y22 32y22
9y22
18y2
8. 12x2y3x2y42 12x2y9x4y8
12
9x24y18
10.
15 31 3 2
15m3
m
m
25m
25
5
12. 2x2y32
4x22y32
2
3xy
3xy2
108x y
6 9
14.
21y 21 y 2
3
3
1
y6
8
2 3
16. 22
5 22 22 5 2
1
254
2
254 2
16
625
2
18.
31 2
2
4x4y6
3xy2
4x41y62
3
4x3y4
3
32 2 3 4 81
Review Exercises for Chapter 5
20. 43x3 4
3x3
26.
22.
15t 5
5
t 53
24t 3 8
4
27x 3
24. 5x2y 42 5
t8
8
1u 0 1 v 2 3 u v
2u 0 v 2
10u 1 v3
5
5
3 1 2
28.
4x8x zz 4
4x8x zz 4
3 1
2
5x 1 y 199
2
2 4
1
5 2 x4 y 8
x4
25y 8
2x 4 3 z 1 1 2
2x7z 22
4x 14 z 4
30. a 4 2a1b 2ab0 2a 4 1b 2 1 2a 3b 2
32. 30,296,000,000 3.0296 1010
34. 2.74 104 0.000274
36. 3 1038 10 7 24 1037
24 10 4
2.4 10 5
38.
1
1
106
3
2
6
6 10 36 10
36
1,000,000
36
250,000
9
40. Standard form: 2x 6 x 5 5x 3 7
Leading coefficient: 2
Degree: 6
42. Standard form: 8x7 x5 2x3 9x
44. Binomial of degree 2 and leading coefficient 7:
7x2 6x
Leading coefficient: 8
Degree: 7
46. 7x 3 x 2 18 x 2 7x 3 18
48. 7 12x 2 8x 3 x 4 6x 3 7x 2 5
x 4 8x 3 6x 3 12x 2 7x 2 7 5
x 2 7x 15
x 4 2x 3 5x 2 2
50. x 2 5 3 6x x 2 5 3 6x
52. 7z2 6z 35z2 2z 7z2 6z 15z2 6z
x 2 6x 5 3
7z2 15z2 6z 6z
x 2 6x 8
8z2
54. 16a3 5a 5a 2a3 1 16a3 5a 5a 10a3 5
16a3 10a3 5a 5a 5
6a3 5
56. 7x4 10x2 4x x3 3x 3x4 5x2 1 7x4 3x4 x3 10x2 5x2 4x 3x 1 4x4 x3 5x2 x 1
6x 1
58. x2 4x
x2 2x 1
60. 10y2
10y 2
3
3
⇒
2
y 4y 9
y 2 4y 9
9y 2 4y 12
200
Chapter 5
Polynomials and Factoring
62. Perimeter 7 8x x 12 6x 2 x 2x 2x x
21x 17
64. Verbal Model:
Equation:
Profit
Revenue Cost
66. 4y2 y 2 16y 2 y 2
16y 3 32y2
P x 1.1x 0.5x 1000
1.1x 0.5x 1000
0.6x 1000
P 5000 0.6 5000 1000
3000 1000
$2000
68. 2y5y2 y 4 10y3 2y2 8y
70. x 6x 9 x2 9x 6x 54
x2 3x 54
72. 4x 12x 5 8x2 20x 2x 5
74. 3y2 24y2 5 3y24y2 5 24y2 5
12y4 15y2 8y2 10
8x2 22x 5
12y4 7y2 10
76. 5s3 4s 34s 5 5s34s 5 4s4s 5 34s 5
20s4 25s3 16s2 20s 12s 15
20s4 25s3 16s2 32s 15
78. 3v 25v 5v3v 2 15v2 10v 15v2 10v
0
80. 2x 3y2 2x2 22x3y 3y2
82. 5x 2y5x 2y 5x2 2y2 25x2 4y2
4x 2 12xy 9y 2
84. m 5 n2 m 52 2m 5n n2
86. Verbal Model:
Area of
Shaded
Region
Equation:
Area 12
m2 2m5 52 2nm 10n n2
m2 10m 25 2mn 10n n2
Area of
Larger Triangle
Area of
Smaller
Triangle
3x 103x 123x2x
32x3x 10 3xx
92x2 15x 3x2
92x2 62x2 15x
3
3
2 x2 15x or 2 xx 10
88. 750 1 r2 750 1 2r r 2 750 1500r 750r 2
or
750r 2 1500r 750
90. 14z3 21 7z3 3
92. a3 4a a a 2 4
Review Exercises for Chapter 5
94. 8y 12y2 24y3 4y 2 3y 6y2
96. u 9vu v vu 9v u 9vu v v
uu 9v
98. y3 4y2 y 4 y3 4y2 y 4
100. x3 7x2 3x 21 x3 7x2 3x 21
y2 y 4 y 4
x2x 7 3x 7
y 4 y2 1
x 7x2 3
y 4 y 1 y 1
102. 16y2 49 4y2 72
104. y 32 16 y 32 42
4y 74y 7
y 3 4 y 3 4
y 7 y 1
108. 64y 3 8 4y3 23
106. t3 125 t3 53
t 5t2 5t 25
4y 216y 2 8y 4
112. 54 2x 3 227 x 3
110. y 4 4y 2 y 2 y 2 4
2x 3 27
y 2 y 2 y 2
2x 3 33
2x 3x 2 3x 9
114. y2 16y 64 y2 28y 82
y 82
116. u2 10uv 25v2 u2 2u5v 5v2
u 5v2
118. x2 12x 32 x 8x 4
120. 5x2 11x 12 5x 4x 3
122. 12x2 13x 14 4x 73x 2
124. 12x 2 7x 1 12x 2 4x 3x 1
4x 3x 1 13x 1
4x 13x 1
126. 3u2 7u 6 3u2 9u 2u 6
128. 3x 2 13x 10 3x 2 15x 2x 10
3u u 3 2u 3
3x x 5 2x 5
3u 2u 3
3x 2x 5
132. x3 3x2 4x 12 x2x 3 4x 3
130. 3b 27b3 3b1 9b2
x 3x2 4
x 3x 2x 2
2
1
1
1
134. x2 3x 9 x2 23 x 3 2
x 1 2
3
201
136. u6 8v6 u23 2v23
u2 2v2u4 2u2v2 4v4
202
Chapter 5
Polynomials and Factoring
138. 7x 2x 5 0
7x 0
140. x 73x 8 0
2x 5 0
x0
x
142. 3x x 82x 7 0
3x 0
3x 8 0
x7
x 83
144. x2 25x 150
x80
x0
x70
5
2
2x 7 0
x2 25x 150 0
7
x 8
x 15x 10 0
x2
x 15 0
x 15
146. 3x4x 7 0
3x 0
148. x 32 25 0
4x 7 0
x0
4x 7
7
x 4
x 10 0
x 10
x 3 5x 3 5 0
x 11x 11 0
x 2x 8 0
x 11 0
x20
x2
3x 2 3x 2 5x 2 0
x x 2 20x 36 0
3x 2 3x 1x 2 0
x x 18x 2 0
x20
x 13
x0
x0
x2
x 2 x 3 5x 3 0
x 2 5x 3 0
x 5 x 5 x 3 0
x 5 0
x 5 0
x30
x 5
x 5
x 3
158. 2x 4 6x 3 50x 2 150x 0
2x 3 x 3 50x x 3 0
2x 3 50xx 3 0
2x x 2 25x 3 0
2x x 5x 5x 3 0
2x 0
x50
x0
x5
x50
x 5
x 18 0
x 18
156. x 3 3x 2 5x 15 0
x30
x 3
x 11
x 8
154. x 3 20x 2 36x 0
3x 1 0
x 11 0
x 11
x80
152. 9x 4 15x 3 6x 2 0
3x2 0
x 2 121 0
150.
x20
x 2
Review Exercises for Chapter 5
160. Verbal Model:
Labels:
First even
integer
Second even
224
integer
First even integer 2n
162. Verbal Model: Area Labels:
2n 2n 2 224
1728 3x 2
4n 8n 7 0
576 x 2
n80
n70
24 x
n 8
n7
18 34 x
2n 14
0 16t 2 45t 664
0 16t 83t 8
0 16t 83
t
reject
8 seconds
432 x 34 x
4n 2 n 56 0
0 16t 2 45t 664
83
16
Equation:
432 34 x 2
2n 2 16
164.
Length x
4n 2 4n 224 0
reject
0t8
8t
3
Width 4 x
Second even integer 2n 2
Equation:
Length
24 inches 18 inches
Width
203