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Chapter 4 Exponents, Polynomials, and Factoring
Section 4.1 Practice Exercises
1. a. exponent
b. 1
c. 1n
b
or 1nb
d. scientific notation
3.
3
3
3 3
ab a b b b
ab ab ab aba a a b b b
a b
5. For example:
2 2 2
3 3 3
5 5x x
xy x y
7. For example:
53
2
42
2
88
8
x xx
9. For example:
0
0
6 1
1 0x x
11. 1 11 3
33 1
13. 2
2
1 15
255
15. 2
2
1 15
255
17.
2
2
1 15
255
19.
3 331 4
4 644 1
21. 44 4
4
23 2 16
2 3 813
23. 3 3 3
3
2 5 5 125
5 2 82
25. 010 1ab 27. 010 10 1 10ab a a
29. 3 5 3 5 8y y y y 31. 88 6 2
6
1313 13 169
13
180
33. 42 2 4 8y y y 35. 4 4
2 4 2 4 2 4 83 3 3 81x x x x
37. 3
3
1pp
39. 10 1310 13 3
3
17 7 7 7
71
343
41. 33 5 2
5 2
1w w ww w
43. 2 52 5 7
7
1a a a aa
45. 1 1 2
1
r r rr
47. 6
6 2 4
2 4
1z z zz z
49. 33 3 2
2 2
1a a a bb b
51. 026 1xyz
53. 4 2 4
2
14
12 2 2
21
164
6516 or
4
55. 2 2
2 2
125
1 11 5
1 51 1
1 2526
1 or 25
57. 2 2 0 22 1 1 3 1
13 2 3 2 4
9 1 4
4 4 412
34
59. 1 2 04 3 2 5 9
15 2 7 4 4
5 9 4
4 4 410 5
4 2
61. 21 12 5
5 1
3 2
22
3 3
1
p q p qp q
p q
qqp p
63. 101 4 10 3 3 7
4 3
77
3 3
48 48 3
32 232
3 1 3
2 2
aba b a b
a b
bba a
Section 4.1 Properties of Integer Exponents and Scientific Notation
181
65.
44 5 2
4 4 44 4 5 2
416 20 8
1616
20 8 20 8
3
3
1
3
1 1 1
81 81
x y z
x y z
x y z
xxy z y z
67. 1 32 6 3 2 6
4 2
4
2
4
2
4 4
4
14
4
m n m n m n
m n
mn
mn
69.
3 22 4
3 22 3 2 2 4
6 3 2 8
6 2 3 8
4 11
1111
4 4
2
2
4
4
4
1 44
p q pq
p q p q
p q p q
p q
p q
qqp p
71.
32 6
2 2
3
6 2 1 3
8 2
8
2
8
2
5 5
5
5
15
5
x xx y x yy y
x y
x y
xy
xy
73.
4 4 442 2 2 2
2 2 223 7 3 7
8 8
6 14
8 6 8 14
2 6
2
6
2
6
8 8
16 16
4096
256
16
16
116
16
a b a b
a b a b
a ba b
a b
a b
ab
ab
75.
36 5
2 4
36 2 5 4
38 9
3 3 38 9
324 27
27
24
27
24
2
3
2
3
2
3
2
3
3
2
27 1
8
27
8
x yx y
x y
x y
x y
x y
yx
yx
182
77.
2 23 00 53 6
6 5
29 5
2 2 29 5
2 18 10
18
10
18
10
2 1
24
1
2
1
2
2
14
4
x y x yx y
x y
x y
x y
xy
xy
79.
24
5
5 3
25 4 5 1 3
25 1 2
2 2 25 1 2
25 2 4
1 2 5 4
3 9
23
6
13
3
13
3
1 3
3
3 3
3 9
27
x yxyx y
xy x y
xy x y
xy x y
xy x y
x y
x y
81. a. 9$8,000,000,000 $8 10 83. a. 112 10 200,000,000,000 b. 63,000,000 3 10 DVDs b. 64 10 0.000004 c. 1314,000,000,000,000 1.4 10 eV c. 111.082 10 108,200,000,000 d.
19
0.000 000 000 000 000 0001 602
1.602 10 J
85. 4 1 4
5
35 10 3.5 10 10
3.5 10
87. 07.0 10 Proper
89. 19 10 Proper 91. 3 83 8
1 5
4
6.5 10 5.2 10 33.8 10
3.38 10 10
3.38 10
93. 6 9
6 9
1 3 4
0.0000024 6,700,000,000
2.4 10 6.7 10
16.08 10
1.608 10 10 1.608 10
95.
2 15
2 15
13
8.5 10 2.5 10
3.4 10
3.4 10
Section 4.2 Addition and Subtraction of Polynomials and Polynomial Functions
183
97. 8 5
8 5
3
900000000 360000
9 10 3.6 10
2.5 10
2.5 10
99.
23 23
1 23
24
23 23
2 6.02 10 12.04 10
1.204 10 10
1.204 10 hydrogen atoms
1 6.02 10 6.02 10 oxygen atoms
101.
6 2
4 2
2,200,000 110
2.2 10 1.1 10
2 10 or 20,000 people per mi
103. 9 9
10
$3.5 10 15 $52.5 10
$5.25 10
105. a. 45 12 540 months b. $20 540 $10,800
c. 5400.06 12
$20 1 1 1 $55,395.4512 0.06
A
107. 5 7 5 7 2 2a a a a ay y y y 109. 3 33 3 1
1
3 3 1 2 4
a a aa
a a a
x xx
x x
111. 2 2 32 2 4 3 3 2 2 4 3 3 6 6
4 3
a a a a a a a a a a aa a
x y x y x y x yx y
Section 4.2 Practice Exercises
1. a. polynomial g. leading; leading coefficient
b. coefficient; n h. greatest
c. 1; 1 i. zero
d. one j. exponents
e. binomial k. polynomial
f. trinomial
3. 1 12 1 4 2 4
0 2 2
2 5 10
10 10
ac a c a c
a c c
5. 5 2 3
4
3.4 10 5.0 10 17 10
1.7 10
184
7. 3 26a a a
leading coefficient: –6
degree: 3
9. 4 23 6 1x x x leading coefficient: 3
degree: 4
11. 2 100t leading coefficient: –1
degree: 2
13. For example: 53x
15. For example: 2 2 1x x 17. For example: 4 26 x x
19. 2 2
2 2
2
4 4 5 6
4 5 4 6
10
m m m m
m m m m
m m
21.
4 3 2 3 2
4 3 3 2 2
4 3 2
3 3 7 2
3 3 7 2
3 2 8 2
x x x x x x
x x x x x x
x x x x
23. 3 2 3 2
3 3 2 2
3 2
1 2 3 11.8 2.7
2 9 2 9
1 3 2 11.8 2.7
2 2 9 91
2 0.99
w w w w w w
w w w w w w
w w w
25. 2 2
2 2
2
9 5 1 8 15
9 8 5 1 15
17 4 14
x y xy x y xy
x y x y xy xy
x y xy
27. 2 2
2 2
2
7 6 1 8 4 2
6 2 7 4 1 8
4 11 7
a a a a
a a a a
a a
29.
3
3 2
3 2
12 6 8
3 5 4
9 5 2 8
x x
x x x
x x x
31. 3 330 30y y 33. 3 34 2 12 4 2 12p p p p
35. 2 2 2 211 11ab a b ab a b 37.
5 2 5 2
5 2 5 2
5 5 2 2
5 2
13 7 5
13 7 5
13 7 5
6 6
z z z z
z z z z
z z z z
z z
Section 4.2 Addition and Subtraction of Polynomials and Polynomial Functions
185
39.
3 2 2 3
3 2 2 3
3 2 3 2
3 3 2 2
3 2
3 3 6 1
3 3 6 1
3 3 6 1
3 3 6 1
2 4 5
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x
41.
3 2 3
3 2 3
3 3 2
3 2
3 3 6 1
3 3 6 1
3 3 6 1
2 3 5
xy x y x xy xy x
xy x y x xy xy x
xy xy x y xy x x
xy x y xy
43.
3 2 3 2
3 2 3 2
3 2
4 6 18 4 6 18
3 7 9 5 3 7 9 5
13 9 13
t t t t
t t t t t t
t t t
45. 2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
2 2
1 1 1 3 2 13 5
5 2 10 10 5 2
1 1 1 3 2 13 5
5 2 10 10 5 2
1 3 1 2 1 13 5
5 10 2 5 10 22 3 5 4 1 5
3 510 10 10 10 10 101 9 3
82 10 5
a ab b a ab b
a ab b a ab b
a a ab ab b b
a a ab ab b b
a ab b
47.
2 2
2 2
2 2
2
8 15 9 5 1
8 15 9 5 1
8 9 5 15 1
6 16
x x x x
x x x x
x x x x
x x
49. 5 3 4 3
5 3 4 3
5 4 3 3
5 4 3
3 2 4 2 7
3 2 4 2 7
3 2 2 4 7
3 4 11
x x x x
x x x x
x x x x
x x x
51.
2 3 2 3
2 3 2 3
3 3 2 2
3 2
8 4 3 8
8 4 3 8
4 8 8 3
4 5
y y y y
y y y y
y y y y
y y
53. 4 4
4 4
4
2 6 9
6 2 9
7 11
r r r r
r r r r
r r
186
55.
2 2
2 2
2 2
2
5 13 3 4 8
5 13 3 4 8
13 4 5 3 8
9 5 11
xy x y x y
xy x y x y
x x xy y y
x xy y
57. 2 2
2 2
2
11 23 7 19
11 7 23 19
18 42
ab b ab b
ab ab b b
ab b
59.
2 3 5 4 6 2
2 3 5 4 6 2
5 4 6 2
4 5 6 2
3 9
p p pp p p
p pp pp
61. 2 2
2 2
2
2 2
5 2 4 1
5 2 4 1
5 2 1
5 2 1 2 6
m m
m m
m
m m
63. 3 3 3
3 3 3
3
6 5 3 2 2 6
6 5 3 2 2 6
7 4 5
x x x x x
x x x x x
x x
65. 2 2 2 2 2 2 2 2
2 2 2
2 2 2
2 2
5 7 2 7 2 5 7 2 7 2
5 5 2 7
5 5 2 7
12 5
ab a b ab ab a b ab ab a b ab ab a b ab
ab a b ab ab a b
ab a b ab ab a b
a b ab ab
67.
3 2 2 3
3 2 2 3
3 2 3 2
3 2 3 2
3 3 2 2
3 2
8 3 5 4 2
8 3 5 4 2
8 3 4 5 2
8 3 4 5 2
8 4 5 3 2
12 6 1
x x x x x x
x x x x x x
x x x x
x x x x
x x x x
x x
69.
2 2 2 2
2 2 2 2
2 2
12 4 1 2 4
4 5 4 5
8 5 4
a b ab ab a b ab ab
a b ab ab a b ab ab
a b ab ab
71.
4 2 4 2
4 3 2 4 3 2
3 2
5 11 6 5 11 6
5 3 5 10 5 5 3 5 10 5
3 16 10 1
x x x x
x x x x x x x x
x x x
Section 4.2 Addition and Subtraction of Polynomials and Polynomial Functions
187
73.
5 4 2
4 3 2
5 4 3 2
2.2 9.1 5.3 7.9
6.4 8.5 10.3
2.2 15.5 8.5 5 7.9
p p p p
p p p
p p p p p
75. 3 3 3
3 3 3
3
2 6 4 5 6
2 6 4 5 6
12 2
P x x x x x x
x x x x x x
x x
77. 225
3h x x
It is a polynomial function. The degree
is 2.
79. 3 2 38 2p x x x
x
It is not a polynomial function. The term
13 3x x and –1 is not a whole
number.
81. 7g x
It is a polynomial function. The degree
is 0.
83. 5M x x x
It is not a polynomial function. The term
x is not of the form nax .
85. a. 4 2 5P x x x
42 2 2 2 5
16 4 5
17
P
87. a. 31 12 4
H x x x
31 10 0 0
2 41 1
0 04 4
H
b. 41 1 2 1 5
1 2 5
8
P
b. 31 12 2 2
2 41 1
4 2 24 4
9
4
H
c. 40 0 2 0 5
0 0 5
5
P
c. 31 12 2 2
2 41
4 24
1 72
4 4
H
d. 41 1 2 1 5
1 2 5
4
P
d. 31 11 1 1
2 41 1 3
12 4 4
H
188
89. Let x = the width of the garden
x + 3 = the length of the garden
2 2 3
2 2 6
4 6
P x x xx xx
91. a. 12 5.40 99
12 5.40 99
6.6 99
P x R x C xx x
x x
x
b. 50 6.6(50) 99
330 99
231
P
The profit will be $231.
93. a. 25.2 40.4 1636D x x x
20 5.2 0 40.4 0 1636
0 0 1636 1636
D
D(0) = 1636 means that at the
beginning of the study, (year 0)the
annual dormitory charge was
$1636.
218 5.2 18 40.4 18 1636
1684.8 727.2 1636 404
D
In 2008, the annual dormitory
charge was $4048.
95. a. 143 6580W t t
0 143 0 6580
6580
5 143 5 6580
715 6580
7295
10 143 10 6580
1430 6580
8010
W
W
W
b. 225 5.2 25 40.4 25 1636
3250 1010 1636 5896
D
The annual dormitory charge will
be $5896 .
b. W(10) = 8010 means that in Year 10,
8010 thousand (8,010,000) women
were due in child support.
97. a. 2
25
16 43.3
x t t
y t t t
2
0 25 0 0
0 16 0 43.3 0 0 0 0
x
y
(0, 0); at t = 0 sec, the position of the
rocket is at the origin.
b.
(25, 27.3) At t = 1 sec, the position
of the rocket is (25, 27.3).
2
1 25 1 25
1 16 1 43.3 1
16 43.3 27.3
x
y
Section 4.3 Multiplication of Polynomials
189
c.
(50, 22.6) At t = 2 sec, the position of the rocket is (50, 22.6).
Section 4.3 Practice Exercises
1. a. distributive c. squares; 2 2a bb. 4 7x d. perfect; 2 22a ab b
3. 2 2
2
2
2
2 3 5 6 4 1 2 3 5 6 4 1
2 3 6 4 4
2 3 6 4 4
6 6
x x x x x x
x x x
x x x
x x
5. a. 4 2 3g x x x
4 21 1 1 3
1 1 3
3
g
7. 4 5 4 5
5 6
7 6 7 6
42
x y xy x x y y
x y
b. 4 22 2 2 3 16 4 3 9g
c. 4 20 0 0 3 0 0 3 3g
9. 6 4 7 4 3 7 8 102.2 5 11a b c ab c a b c 11. 1 1 1 2 32 3 2 3
5 5 5 5 5a a a
13.
3 2 2 3 2
3 2 2 3 3 2 2 3 2
5 5 4 4 3 3
2 3 4
2 2 3 2 4
2 6 8
m n m n mn n
m n m n m n mn m n n
m n m n m n
15. 2 2 2
2 2 2 3
1 2 1 26 6 6
2 3 2 3
3 4
xy x xy xy x xy xy
x y x y
2
2 25 2
50
2 16 2 43.3 2
64 86.6
22.6
x
y
190
17. 2 2
2 2
2
2 2
2 2
2
x y x yx x x y y x y y
x xy xy y
x xy y
19.
2
2
6 1 5 2
6 5 6 2 1 5 1 2
30 12 5 2
12 28 5
x xx x x x
x x x
x x
21.
2 2
2 2 2 2
4 2 2
4 2
12 2 3
2 3 12 2 12 3
2 3 24 36
2 21 36
y y
y y y y
y y y
y y
23.
2 2
2 2
5 3 5 2
5 5 5 2 3 5 3 2
25 10 15 6
25 5 6
s t s ts s s t t s t t
s st st t
s st t
25.
2
2 2
3 2
10 5 3
5 3 10 5 10 3
5 3 50 30
n n
n n n n
n n n
27.
2 2
2 2
1.3 4 2.5 7
1.3 2.5 1.3 7 4 2.5 4 7
3.25 9.1 10 28
3.25 0.9 28
a b a ba a a b b a b b
a ab ab b
a ab b
29.
2 2
2 2 2 2
3 2 2 2 2 3
3 2 2 3
2 3 2
2 3 2 2 2 3 2
6 4 2 3 2
6 7 4
x y x xy y
x x x xy x y y x y xy y y
x x y xy x y xy y
x x y xy y
31. 2 2 2
3 2 2
3
7 7 49 7 49 7 7 7 7 49
7 49 7 49 343
343
x x x x x x x x x x
x x x x x
x
33.
3 2 2 3
3 2 2 3 3 2 2 3
4 3 2 2 3 3 2 2 3 4
4 3 2 2 3 4
4 4
4 4 4 4 4 4
4 16 4 4 4
4 17 8 5
a b a a b ab b
a a a a b a ab a b b a b a b b ab b b
a a b a b ab a b a b ab b
a a b a b ab b
Section 4.3 Multiplication of Polynomials
191
35.
2 2 2
2 2 2
12 6
2
1 1 16 2 2 6 2 6
2 2 21 1
3 2 12 2 62 21 1
12 82 2
a b c a b c
a a a b a c b a b b b c c a c b c c
a ab ac ab b bc ac bc c
a ab ac b bc c
37. 2 2 2
3 2 2
3 2
2 1 3 5 3 5 2 3 2 5 1 3 1 5
3 5 6 10 3 5
3 11 7 5
x x x x x x x x x x
x x x x x
x x x
39.
2 2
1 110 15
5 2
1 1 1 115 10 10 15
5 2 5 2
1 13 5 150 8 150
10 10
y y
y y y y
y y y y y
41. 2 2
2
8 8 8
64
a a a
a
43. 2 2 23 1 3 1 3 1 9 1p p p p 45. 22
2
1 1 1
3 3 3
1
9
x x x
x
47. 2 2
2 2
3 3 3
9
h k h k h k
h k
49. 2 2 2
2 2
3 3 2 3
9 6
h k h h k k
h hk k
51. 2 2 2
2
7 2 7 7
14 49
t t t
t t
53. 2 22
2 2
3 2 3 3
6 9
u v u u v v
u uv v
55.
2 22
2 2
1 1 12
6 6 6
1 1
3 36
h k h h k k
h hk k
57. 2 22 3 2 3 2 3
4 6
2 2 2
4
z w z w z w
z w
192
59. 2 2 22 2 2 4 2 25 3 5 2 5 3 3 25 30 9x y x x y y x x y y
61. a. When two conjugates are multiplied,
the resulting binomial is a difference
of squares.
2
2
( 5 4)(5 4)
25 20 20 16
16 25
x xx x x
x
Since 2( 5 4)(5 4) 16 25x x x is a difference of squares, the
binomials are conjugates.
63. a. 2 2A B A B A B
b. 2 2
2 2 22
x y B x y B
x y B
x xy y B
Both are examples of multiplying
conjugates to get a difference of
squares.
b. When two conjugates are multiplied,
the resulting binomial is a difference
of squares.
2
2
( 5 4)(5 4)
25 20 20 16
25 40 16
x xx x xx x
Since
2( 5 4)(5 4) 25 40 16x x x x is not a difference of squares, the
binomials are not conjugates.
65. 2 2
2 2
2 2 2
2 4
w v w v w v
w wv v
67.
22
2 2
2 2
2 2 2
4 2
4 2
x y x y x y
x xy y
x xy y
69.
2 2
2 2 2
2 2
3 4 3 4
3 4
3 2 3 4 4
9 24 16
a b a b
a b
a a b
a a b
71. Write 3 2 as x y x y x y .
Square the binomial and then use the
distributive property to multiply the
resulting trinomial by the remaining
factor of x y .
Section 4.3 Multiplication of Polynomials
193
73.
3 2
2 2
2 2 2 2
3 2 2 2 2 3
3 2 2 3
2 2 2
4 4 2
4 2 4 4 2 4 2
8 4 8 4 2
8 12 6
x y x y x y
x xy y x yx x x y xy x xy y y x y yx x y x y xy xy yx x y xy y
75.
3 2
2 2
2 2 2 2
3 2 2 2 2 3
3 2 2 3
4 4 4
16 8 4
16 4 16 8 4 8 4
64 16 32 8 4
64 48 12
a b a b a b
a ab b a ba a a b ab a ab b b a b ba a b a b ab ab ba a b ab b
77. Multiply the first two binomials and
simplify.
Then multiply the resulting trinomial and
the third binomial, using the distributive
property.
79.
2
2
2 2
2 2
2 2 2 2
4 3 2
2 5 3 1
2 3 1 5 3 5 1
2 3 15 5
2 3 16 5
2 3 2 16 2 5
6 32 10
a a a
a a a a a
a a a a
a a a
a a a a a
a a a
81.
2
2 2
3 2
3 3 5
9 5
5 9 9 5
5 9 45
x x x
x x
x x x x
x x x
83.
6 3
6 3
3 32
2 4 2 2
128 54
2 64 27
2 4 3
2 4 3 16 12 9
p q
p q
p q
p q p p q q
85. 2 2 2 2
2 2
2
1 2 3 2 1 4 12 9
2 1 4 12 9
3 10 8
y y y y y y
y y y y
y y
87. 2r t 89. 2 3x y
194
91. The sum of the cube of p and the square
of q.
93. The product of x and the square of y.
95. Let x = the width of the walk
2x + 20 = length of garden and walk
2x + 15 = width of garden and walk
2
2
2 20 2 15
2 2 2 15 20 2 20 15
4 30 40 300
4 70 300
A x x xx x x x
x x x
x x
97. a. Let x = the length of a side of the
square
8 – 2x = length and width of
base
x = the height of the box
2
3 2
8 2 8 2
64 32 4
4 32 64
V x x x x
x x x
x x x
b. 3 2
3
1 4 1 32 1 64 1
4 32 64
36 in
V
99. 2 2 2
2
2 2 2 2
4 4
x x x
x x
101. 2 2
2
2 2 2
4
x x xx
103. 2 2
2
12 6 3 3 3
23
9
x x x x
xx
105.
2
2 2
3 2
3 3 10 3 3 10
3 3 3 10
9 30
x x x x xx x xx x
107. 2 22 2 2
2 2 2
2
( ) 3( ) 5 3 5 2 3 3 5 3 5
2 3 3 3 5 5
2 3
(2 3)
2 3
x h x h x x x xh h x h x xh h
x x xh h x x hh
xh h hh
h x hh
x h
Section 4.4 Division of Polynomials
195
109. Multiply 2 22 2x x by squaring
the binomials.
Then multiply the resulting trinomials
using the distributive property.
111. 5 6x Check:
2
2
2 3 5 6
2 5 2 6 3 5 3 6
10 12 15 18
10 27 18
x xx x x x
x x x
x x
113. 2 1y Check:
2
2
4 3 2 1
4 2 4 1 3 2 3 1
8 4 6 3
8 2 3
y yy y y y
y y y
y y
Section 4.4 Practice Exercises
1. a. division; quotient; remainder b. Synthetic
3. a. 10 5 10 5
4 11
a b a b a b a ba b
5. a. 2 2
2 2
2
6 2
6 2
7 2
x x x xx x x xx
b.
2 2
2 2
10 5
5 10 5 10
5 50 10
5 49 10
a b a ba a a b b a b ba ab ab ba ab b
b.
2 2
2 2 2 2
2
4 3 2 3 2
4 3 2
6 2
6 2
6 2
6 2 6 2
6 5 2
x x x x
x x x x x
x x x x xx x x x x xx x x x
7. For example:
2 2 2
2
5 1 5 2 5 1 1
25 10 1
y y y
y y
9. 4 2 4 2
3
16 4 20 16 4 20
4 4 4 4
4 5
t t t t t tt t t t
t t
196
11. 2 3
2 3
2
36 24 6 3
36 24 6
3 3 3
12 8 2
y y y y
y y yy y y
y y
13. 3 2 2 3
3 2 2 3
2 2
4 12 4 4
4 12 4
4 4 4
3
x y x y xy xy
x y x y xyxy xy xy
x xy y
15. 4 3 2 2
4 3 2
2 2 2
2
8 12 32 4
8 12 32
4 4 4
2 3 8
y y y y
y y yy y y
y y
17. 4 3 2
4 3 2
3 2
3 6 2 6
3 6 2
6 6 6 6
1 1 1
2 3 6
p p p p p
p p p pp p p p
p p p
19. 3 2
3 2
2
5 5
5 5
55 1
a a a a
a a aa a a a
a aa
21. 3 5 2 4 2
4
3 5 2 4 2
4 4 4
2
2
6 8 10
2
6 8 10
2 2 2
53 4
s t s t stst
s t s t stst st st
s t st
23. 4 7 5 6 3 2
4 7 5 6 3
2 2 2 2
2 6 3 5
2
8 9 11 4
8 9 11 4
48 9 11
p q p q p q p q
p q p q p qp q p q p q p q
p q p q pp q
Section 4.4 Division of Polynomials
197
25. a.
2
3 2
3 2
2
2
2 3 1
2 2 7 5 1
2 4
3 5
3 6
1
2
3
x xx x x x
x x
x x
x x
xx
Divisior: ( 2)x Quotient:
2(2 3 1)x x Remainder: (–3)
b. Multiply the quotient and divisor; then
add the remainder.
The result should equal the dividend.
27.
2
2
7
4 11 19
4
7 19
7 28
9
xx x x
x x
xx
Solution: 9
74
xx
Check:
2
2
7 11 28 9
11 19
4 9x x x
x x
x
29.
2
3 2
3 2
2
2
3 2 2
3 3 7 4 3
3 9
2 4
2 6
2 3
2 6
9
y y
y y y y
y y
y y
y y
yy
Solution: 2 9
33
2 2yy
y
Check:
2
3 2 2
3 2
33 2 2 9
3 2 2 9 6 6 9
3 7 4 3
yy y
y y y y yy y y
198
31.
2
2
4 11
3 11 12 77 121
12 44
33 121
33 121
0
a
a a a
a a
aa
Solution: 4 11a
Check:
2
2
3 11 4 11 0
12 33 44 121
12 77 121
a aa a aa a
33.
2
2
6 5
3 4 18 9 20
18 24
15 20
15 20
0
y
y y y
y y
yy
Solution: 6 5y
Check:
2
2
3 4 6 5 0
18 15 24 20
18 9 20
y yy y yy y
35.
2
3
3 2
2
2
6 4 5
3 2 18 7 12
18 12
12 7
12 8
15 12
15 10
22
x xx x x
x x
x x
x x
xx
Solution: 2 226 4 5
3 2x x
x
Check:
2
3 2 2
3
3 2 6 4 5 22
18 12 15 12 8 10 22
18 7 12
x x x
x x x x x
x x
37.
2
3
3 2
2
2
4 2 1
2 1 8 1
8 4
4
4 2
2 1
2 1
0
a aa a
a a
a
a a
aa
Solution: 24 2 1a a
Check:
2
3 2 2
3
42 1 2 1 0
8 4 2 4 2 1
8 1
aa a
a a a a a
a
Section 4.4 Division of Polynomials
199
39.
2
2 4 3 2
4 3 2
3
3 2
2
2
2 2
1 4 2
2 4
2 2 2
2 2 2
2 2 2
0
x xx x x x x x
x x x
x x
x x x
x x
x x
Solution: 2 2 2x x
Check:
2 2
4 3 2 3 2
2
4 3 2
1 2 2 0
2 2 2 2
2 2
4 2
x x x x
x x x x x x
x x
x x x x
41.
2
2 4 3
4 2
3 2
3
2
2
2 5
5 2 10 25
5
2 5 10
2 10
5 25
5 25
0
x xx x x x
x x
x x x
x x
x
x
Solution: 2 2 5x x
Check:
2 2
4 3 2 2
4 3
5 2 5 0
2 5 5 10 25
2 10 25
x x x
x x x x x
x x x
43.
2
2 4 2
4 2
2
2
1
2 3 10
2
10
2
8
xx x x
x x
x
x
Solution: 2
2
81
2x
x
Check: 2 2
4 2 2
4 2
2 1 8
2 2 8
3 10
x x
x x x
x x
45.
3 2
4
4 3
3
3 2
2
2
2 4 8
2 16
2
2
2 4
4
4 8
8 16
8 16
0
n n nn n
n n
n
n n
n
n n
nn
Solution: 3 22 4 8n n n
200
Check:
3 2
4 3 2 3 2
4
2 2 4 8 0
2 4 8 2 4
8 16
16
n n n n
n n n n n nn
n
47. The divisor must be of the form x r . 49. No, the divisor is not of the form x r .
51. a.b.c.
Divisor: 5xQuotient: 2 3 11x x
Remainder: 58
53. 8 1 2 48
8 48
1 6 0
Quotient: 6xCheck:
2
2
8 6 0 6 8 48
2 48
x x x x x
x x
55. 1 1 3 4
1 4
1 4 0
Quotient: 4t Check:
2
2
1 4 0 4 4
3 4
t t t t t
t t
57. 1 5 5 1
5 10
5 10 11
Quotient: 11
5 101
yy
Check:
2
2
5
1 5 10 11
10 5 10 11
5 5 1
y y
y y y
y y
59. 3 3 7 4 3
9 6 6
3 2 2 3
Quotient: 2 3
3 2 23
y yy
61. 2 1 3 0 4
2 2 4
1 1 2 0
Quotient: 2 2x x
Section 4.4 Division of Polynomials
201
Check:
2
3 2 2
3 2
3
3 3 2 2 3
2 2 9 6 6 3
3 7 4 3
y y y
y y y y y
y y y
Check:
2
3 2 2
3 2
2 2 0
2 2 2 4
3 4
x x x
x x x x x
x x
63. 2 1 0 0 0 0 32
2 4 8 16 32
1 2 4 8 16 0
Quotient: 4 3 22 4 8 16a a a a
Check:
4 3 2
5 4 3 2
4 3 2
5
2 2 4 8 16 0
2 4 8 16
2 4 8 16 32
32
a a a a a
a a a a a
a a a a
a
65. 6 1 0 0 216
6 36 216
1 6 36 0
Quotient: 2 6 36x x
Check:
2
3 2 2
3
6 6 36 0
6 36 6 36 216
216
x x x
x x x x x
x
67. 26 7 1 3
3
4 2 2
6 3 3 5
Quotient: 2
23
56 3 3t t
t
Check:
2
23
2
23
3 2 2
3 2
2 5(6 3 3)
3
2 2 5(6 3 3)
3 3
6 3 3 4 2 2 5
6 7 3
t t tt
t t t tt
t t t t t
t t t
69. 14 0 1 6 3
2
2 1 0 3
4 2 0 6 0
Quotient: 3 24 2 6w w
Check:
3 2
4 3 3 2
4 2
14 2 6 0
2
4 2 6 2 3
4 6 3
w w w
w w w w w
w w w
202
71. 4 1 8 3 2
4 16 52
1 4 13 54
Quotient: 2 544 13
4x x
x
73. 2
2
22 11 33 11
22 11 33 32 1
11 11 11
x x x
x x xx x x x
75.
2 3 2
3 2
2
2
4 3
3 2 5 12 17 30 10
12 8 20
9 10 10
9 6 15
4 5
y
y y y y y
y y y
y y
y y
y
Quotient: 2
4 54 3
3 2 5
yyy y
77.
2
2 4 3
4 2
3 2
3
2
2
2 3 1
2 1 4 6 3 1
4 2
6 2 3
6 3
2 1
2 1
0
x xx x x x
x x
x x x
x x
x
x
Quotient: 22 3 1x x
79. 11 10 8 4 8
11 10 8 4
8 8 8 8
3 2
4
16 32 8 40 8
16 32 8 40
8 8 8 85
2 4 1
k k k k k
k k k kk k k k
k kk
81. 3 2 2
3 2
2 2 2
5 9 10 5
5 9 10
5 5 59 2
5
x x x x
x x xx x x
xx
83. a.
3 24 4 4 10 4 8 4 20
4 64 10 16 32 20
256 160 32 20
84
P
b. 4 4 10 8 20
16 24 64
4 6 16 84
Quotient: 2 844 6 16
4x x
x
c. The values are the same.
85. P r equals the remainder of P x x r .
Problem Recognition Exercises: Operations on Polynomials
203
87. a. 1 8 13 5
8 5
8 5 0
Quotient: 8 5x
b. Yes
Yes
Problem Recognition Exercises
1. a. 2 2 2
2
3 1 3 2 3 1 1
9 6 1
x x x
x x
3. a. 2 24 8 10 4 8 10
2 2 2 25
2 4
x x x xx x x x
xx
b 2 2
2
3 1 3 1 3 1
9 1
x x x
x
b.
2
2
2 5
2 1 4 8 10
4 2
10 10
10 5
5
xx x x
x x
x
x
Solution: 5
2 52 1
xx
c. 3 1 3 1 3 1 3 1
2
x x x x
c. 1 4 8 10
4 12
4 12 2
Quotient: 2
4 121
xx
5. a. 2
2 2
5 5 5
25 5
30
p p p
p p
b. 22 2
5 5 5
25 10 25
10 50
p p p
p p pp
c. 2
2 2
5 5 25
25 25 0
p p p
p p
7. 2 2 2 2
2
5 6 2 3 7 3 5 6 2 3 7 3
2 1
t t t t t t t t
t t
204
9. 2 2
2
6 5 6 5 6 5
36 25
z z z
z
11. 2
2
3 4 2 1
3 2 3 1 4 2 4 1
6 3 8 4
6 11 4
b bb b b b
b b b
b b
13. 3 2 2
3 2 2
3 2
4 9 12 2 6
4 9 12 2 6
6 8 3
t t t t t t
t t t t t t
t t t
15.
2
2 2
2
2
4 4 9
2 4 4 4 9
8 16 4 9
4 25
k k
k k k
k k k
k k
17.
2
3 2 2
3 2 3
3 2
2 6 3 3 2 3 2
2 12 6 9 4
2 12 6 9 4
7 12 2
t t t t t t
t t t t t
t t t t t
t t t
19. 3 2 3 2
3 2 3 2
3 2
1 1 2 1 15
4 6 3 3 5
3 1 8 2 15
12 6 12 6 511 1 1
512 2 5
p p p p p
p p p p p
p p p
21. 2 2 22 2 2
4 2 2
6 4 6 2 6 4 4
36 48 16
a b a a b b
a a b b
23. 2
2
2
3 2 8
6 9 2 16
8 7
m m
m m m
m m
25. 2 2 2 2 2 2
4 3 2 3 2 2
4 3 2
6 7 2 4 3 2 4 3 6 2 4 3 7 2 4 3
2 4 3 12 24 18 14 28 21
2 8 13 46 21
m m m m m m m m m m m m
m m m m m m m m
m m m m
27. 2 22
2 2
5 5 2 5
25 10 10 2
a b a b a b
a b a ab b
29.
2 2
2 2 2 2
2 2 2 2
2 2
2 2
4
x y x y
x xy y x xy y
x xy y x xy yxy
31. 2 21 1 1 1 1 1 1 1 1 1 1
2 3 4 2 8 4 12 6 8 3 6x x x x x x x
Section 4.5 Greatest Common Factor and Factoring by Grouping
205
Section 4.5 Practice Exercises
1. a. product c. greatest common factor
b. greatest common factor d. grouping
3. 4 3 4 2
4 3 4 2
4 3 2
7 5 9 2 6 3
7 5 9 2 6 3
9 5 6 6
t t t t t t
t t t t t t
t t t t
5. 2 2
4 3 2 2
4 3 2
5 3 2
5 5 10 3 3 6
5 5 7 3 6
y y y
y y y y y
y y y y
7. 3 2 3 2
2
6 12 2 6 12 2
2 2 2 2
3 6 1
v v v v v vv v v v
v v
9.
3 12 3 3 4
3 4
x xx
11. 26 4 2 3 2 2 2 3 2z z z z z z z 13. 6 5 54 4 4 4 1 4 1p p p p p p p
15.
4 2 2 2 2
2 2
12 36 12 12 3
12 3
x x x x x
x x
17.
29 27 9 9 3
9 3
st t t st tt st
19.
4 3 3 4 2 5
2 3 2 2 3 2 3 2
2 3 2 2
9 27 18
9 9 3 9 2
9 3 2
a b a b a b
a b a a b ab a b b
a b a ab b
21.
2 210 15 5
5 2 5 3 5 1
5 2 3 1
x y xy xyxy x xy y xyxy x y
23.
2 2
2
2
13 11 12
13 11 12
13 11 12
b a b ab
b b b a b a
b b a a
25. 2 210 7 1 10 7x x x x
27.
3 2
2
2
12 6 3
3 4 3 2 3 1
3 4 2 1
x y x y xy
xy x xy x xy
xy x x
29.
3 2
2
2
2 11 3
2 11 3
2 11 3
t t t
t t t t t
t t t
206
31.
2 3 2 5 3 2
3 2 2 5
a z b z bz b a
33. 2 22 2 3 2 3 2 3 2 1x x x x x
35. 2 2 22 1 3 2 1 2 1 3y x x x y 37.
2 2
2 2
2
3 2 6 2
3 2 2 2
3 2 2
y x x
y x x
x y
39. For example: 3 2 43 6 12x x x 41. For example: 6 c d y c d
43. a.
2 6 3
2 3 2
2 3
ax ay bx bya x y b x y
x y a b
45.
3 2 2
2
4 3 12 4 3 4
4 3
y y y y y y
y y
b.
210 5 6 3
5 2 1 3 2 1
2 1 5 3
w w bw bw w b ww w b
c. In part (b), –3b was factored out so
that the signs in the last two terms
were changed. The resulting
binomial factor matches the
binomial factor in the first two
terms.
47.
6 42 7 6 7 7
7 6
p pq q p q pp q
49.
2 2 3 3
2 3
2 3
mx nx my nyx m n y m nm n x y
51.
10 15 8 12
5 2 3 4 2 3
2 3 5 4
ax ay bx bya x y b x yx y a b
53.
3 2 2
2
3 3 1 3 1
1 3
x x x x x x
x x
Section 4.5 Greatest Common Factor and Factoring by Grouping
207
55.
2 26 18 30 90
6 3 5 15
6 3 5 3
6 3 5
p q pq p pp pq q p
p q p pp p q
57.
3 2
3 2
2
2
100 300 200 600
100 3 2 6
100 3 2 3
100 3 2
x x x
x x x
x x x
x x
59.
6 2 3
6 2 3
2 3 3
3 2
ax by bx ayax bx ay byx a b y a ba b x y
61.
4 3 12
4 12 3
4 3 4
4 3
a b aba ab b
a b bb a
63. 3 27 21 5 10y y y cannot be
factored.
65. It is not possible to get a common
binomial factor regardless of the order of
the terms.
67.
U Av AcwU A v cw
U Av cw
69.
or
ay bx cybx cy aybx y c a
bx bxy yc a a c
71.
22
2 1
A w wA w w
The length of the rectangle is 2w + 1.
73.
4 5
4
4
4
3 6 3
3 1 6 3
3 1 6 18
3 6 19
a a
a a
a a
a a
75.
3 2
2
2
2
2
2
24 3 5 30 3 5
6 3 5 4 3 5 5
6 3 5 12 20 5
6 3 5 12 15
6 3 5 3 4 5
18 3 5 4 5
x x
x x
x x
x x
x x
x x
77.
24 4
4 4 1
4 3
t t
t tt t
208
79.
3 2 22 3 2
22
22
15 2 1 5 2 1 5 2 1 3 2 1
5 2 1 6 3
5 2 1 7 3
w w w w w w w w
w w w w
w w w
Section 4.6 Practice Exercises
1. a. positive b. opposite
c. 2
2
2 3 4 2 8 3 12
2 5 12
x x x x x
x x
2
2
4 2 3 2 3 8 12
2 5 12
x x x x x
x x
Both are correct.
d.
2 2
2
6 4 10 2 3 2 5
2 3 3 5 5
2 3 1 5 1
2 3 5 1
x x x x
x x x
x x xx x
e. 2( )a b ; 2( )a b
3.
2 7 11 3 5 15 2 4 7
2 4 7 3 4 8
36 12 6
6 6 2 1
c d e c d e c d e
c d e d e cde
5. 2 3 3 3 2 1x a b a b a b x
7.
2 2 33 66
2 33 2
2 33
wz wz az awz z a zz wz a
9.
2 212 32 4 8 32
4 8 4
4 8
b b b b bb b bb b
11.
2 210 24 12 2 24
12 2 12
12 2
y y y y yy y yy y
13.
2 213 30 10 3 30
10 3 10
10 3
x x x x xx x xx x
15.
2
2
6 16
8 2 16
8 2 8
8 2
c c
c c cc c cc c
17.
2
2
2 7 15
2 10 3 15
2 5 3 5
5 2 3
x x
x x xx x xx x
Section 4.6 Factoring Trinomials
209
19.
2 2
2
6 5 6 5
6 6 5 5
6 1 5 1
1 6 5
a a a a
a a aa a aa a
21.
2 2 2 26 3 2 6
3 2 3
3 2
s st t s st st ts s t t s ts t s t
23.
2 2
2
3 60 108 3 20 36
3 18 2 36
3 18 2 18
3 18 2
x x x x
x x x
x x xx x
25.
2 2
2
2 2 24 2 12
2 4 3 12
2 4 3 4
2 4 3
c c c c
c c c
c c cc c
27.
2 2 2 2
2 2
2 8 10 2 4 5
2 5 5
2 5 5
2 5
x xy y x xy y
x xy xy y
x x y y x yx y x y
29. 233 18 2t t Since there are not two factors of 66
whose sum is –18, the polynomial is
prime.
31.
2 2 2 23 14 15 3 9 5 15
3 3 5 3
3 3 5
x xy y x xy xy yx x y y x yx y x y
33.
3 2 2 3 2 2
2 2
2
5 30 45 5 6 9
5 3 3 9
5 3 3 3
5 3 3
5 3
u v u v uv uv u uv v
uv u uv uv v
uv u u v v u vuv u v u v
uv u v
35.
3 2 2
2
5 14 5 14
7 2 14
7 2 7
7 2
x x x x x x
x x x x
x x x xx x x
37.
2 2
2
23 5 10 10 23 5
10 25 2 5
5 2 5 2 5
2 5 5 1
z z z z
z z zz z zz z
39. 2 2 15b b Since there are not two factors of 15
whose sum is 2, the polynomial is
prime.
41.
2 2
2
2 12 80 2 6 40
2 10 4 40
2 10 4 10
2 10 4
t t t t
t t t
t t tt t
210
43.
2 214 13 12 14 21 8 12
7 2 3 4 2 3
2 3 7 4
a a a a aa a aa a
45.
2 2
2
6 22 12 2 3 11 6
2 3 9 2 6
2 3 3 2 3
2 3 3 2
a b ab b b a a
b a a a
b a a ab a a
47. a. 2
2
5 5 5 5 25
10 25
x x x x x
x x
49. a. 2 2
2 2
3 2 3 2
9 6 6 4
9 12 4
x y x y
x xy xy y
x xy y
b. 22 10 25 5x x x b. 22 29 12 4 3 2x xy y x y
51.
22 2
2
9 25 3 2 3 5 5
9 30 25
x x x
x x
53.
24 2 2 2 2
4 2 2
64 8 2 8
64 16
z t z z t t
z z t t
55.
2 2 2
2
8 16 2 4 4
4
y y y y
y
57.
22 2
2
64 80 25 8 2 8 5 5
8 5
m m m m
m
59. 2 2 25 9 2 3 3w w w w
Not a perfect square trinomial.
61.
2 2
2 2
2
9 30 25
3 2 3 5 5
3 5
a ab b
a a b b
a b
63. 2 2 2 216 80 20 4 4 20 5t tv v t tv v
Not a perfect square trinomial.
65.
4 2 4 2
22 2 2
22
5 20 20 5 4 4
5 2 2 2
5 2
b b b b
b b
b
67. a.
2 2 2
2
10 25 2 5 5
5
u u u u
u
69. a.
2 211 26 13 2 26
13 2 13
13 2
u u u u uu u uu u
Section 4.6 Factoring Trinomials
211
b.
24 2 2 2
2
22
22
10 25 10 25
Let
10 25 5
5
x x x x
u x
u u u
x
b.
26 3 3 3
3
2
3 3
11 26 11 26
Let
11 26 13 2
13 2
w w w w
u w
u u u u
w w
c.
2
22
2
2
1 10 1 25
Let 1
10 25 5
1 5
4
a au a
u u u
a
a
c.
2
2
4 11 4 26
Let 4
11 26 13 2
4 13 4 2
9 6
y yu y
u u u u
y yy y
71.
2
2 2
3 1 3 1 6
Let 3 1
6 3 2 6
3 2 3
3 2
3 1 3 3 1 2
3 4 3 1
x xu x
u u u u uu u uu u
x xx x
73.
2
2 2
2 5 9 5 4
Let 5
2 9 4 2 8 4
2 4 4
4 2 1
5 4 2 5 1
1 2 10 1
1 2 9
x xu x
u u u u uu u uu u
x xx xx x
75.
2
2 2
3 4 5 4 2
Let 4
3 5 2 3 6 2
3 2 2
2 3 1
4 2 3 4 1
6 3 12 1
6 3 11
y yu y
u u u u uu u uu u
y yy yy y
77.
6 3
3
2 2
3 3
3 11 6
Let
3 11 6 3 9 2 6
3 3 2 3
3 3 2
3 3 2
y y
u y
u u u u uu u uu u
y y
212
79.
4 2
2
2 2
2 2
4 5 1
Let
4 5 1 4 4 1
4 1 1
1 4 1
1 4 1
p p
u p
u u u u uu u uu u
p p
81.
4 2
2
2 2
2 2
15 36
Let
15 36 12 3 36
12 3 12
12 3
12 3
x x
u x
u u u u uu u uu u
x x
83. The factorization 2 1 2 4y y is not
factored completely because the factor
2 4y has a greatest common factor of
2.
85.
24 2 2 2 2
22
12 36 2 6 6
6
w w w w
w
87.
22 2
2
81 90 25 9 2 9 5 5
9 5
w w w w
w
89.
3 6 3 6
3 2
x a b a b a b xa b x
91.
2 2 2 2 3
2
12 4 6
2 6 2 3
a bc ab c abc
abc a b c
93.
3 2 2
2
20 74 60 2 10 37 30
2 10 25 12 30
2 5 2 5 6 2 5
2 2 5 5 6
x x x x x x
x x x x
x x x xx x x
95. 22 9 4y y Since there are not two factors of –8 whose sum is –9, the polynomial is prime.
97.
22 2
2
2 2
2 2
2 2 2 2
2 5 5 15
Let 5
2 15 2 6 5 15 2 3 5 3
3 2 5 5 3 2 5 5
5 3 2 10 5 2 2 15
w w
u w
u u u u u u u u
u u w w
w w w w
Section 4.7 Factoring Binomials
213
99.
2 21 4 3 1 3 3
1 3 1 3
1 3 1 or 3 1 1
d d d d dd d dd d d d
101.
25 2 10
5 2 5
5 2
ax a bx aba x a b x ax a a b
103.
2 2 2 2
2 2
8 24 224 8 3 28
8 7 4 28
8 7 4 7
8 7 4
z zw w z zw w
z zw zw w
z z w w z wz w z w
105.
5 5 5
5
ay ax cy cx a y x c y xy x a c
107.
2
2
3 14 8
3 12 2 8
3 4 2 4
4 3 2
g x x x
x x xx x xx x
109.
2
2 2
2
20 100
2 10 10
10
n t t t
t t
t
111.
4 3 2
2 2
2 2
2
2
6 8
6 8
4 2 8
4 2 4
4 2
Q x x x x
x x x
x x x x
x x x x
x x x
113.
3 2
2
2
4 2 8
4 2 4
4 2
k a a a a
a a a
a a
Section 4.7 Practice Exercises
1. a.b.c.d.
difference; ( )( )a b a b sum
is not
square
e. f. g. h.
sum; cubes
difference; cubes
;a b 2 2a ab b
;a b 2 2a ab b
3.
22 2
2
4 20 25 2 2 2 5 5
2 5
x x x x
x
5.
10 6 5 3 2 5 3 5 3
5 3 2 1
x xy y x y yy x
214
7.
2 2
2
32 28 4 4 8 7 1
4 8 8 1
4 8 1 1
4 1 8 1
p p p p
p p p
p p pp p
9. Look for a binomial of the form
2 2a b ; 2 2a b a b a b
11.
2 2 29 3
3 3
x xx x
13.
22 216 49 4 7
4 7 4 7
w ww w
15.
2 2 2 2
2 2
8 162 2 4 81
2 2 9
2 2 9 2 9
a b a b
a b
a b a b
17. 225 1u Prime
19.
4 4
2 2
2
2 32 2 16
2 4 4
2 4 2 2
a a
a a
a a a
21.
26 2 3
3 3
49 7
7 7
k k
k k
23.
3 2 2
2
2 2
16 16 1 16 1
1 16
1 4
1 4 4
x x x x x x
x x
x x
x x x
25.
3 2 2
2
2 2
4 12 3 4 3 3
3 4 1
3 2 1
3 2 1 2 1
x x x x x x
x x
x x
x x x
27.
3 2
2
2
2 2
9 7 36 28
9 7 4 9 7
9 7 4
9 7 2
9 7 2 2
y y y
y y y
y y
y y
y y y
29.
2 2 2 2
2 2
49 28 4 49 28 4
7 2
7 2 7 2
x x y x x y
x yx y x y
Section 4.7 Factoring Binomials
215
31.
2 2
2 2
22
9 6 1
9 6 1
3 1
3 1 3 1
3 1 3 1
w n n
w n n
w n
w n w nw n w n
33.
4 2 4
4 2 4
2 22 2
2 2 2 2
10 25
10 25
5
5 5
p p t
p p t
p t
p t p t
35.
4 4 2
4 4 2
2 22 2
2 2 2 2
2 2 2 2
9 4 20 25
9 4 20 25
3 2 5
3 2 5 3 2 5
3 2 5 3 2 5
u v v
u v v
u v
u v u v
u v u v
37. Look for a binomial of the form 3 3a b ;
3 3 2 2a b a b a ab b
39.
33 3
2 2
2
8 1 2 1
2 1 2 2 1 1
2 1 4 2 1
x x
x x x
x x x
Check:
2
3 2 2
3
2 1 4 2 1
8 4 2 4 2 1
8 1
x x x
x x x x x
x
41.
33 3
2 2
2
125 27 5 3
5 3 5 5 3 3
5 3 25 15 9
c c
c c c
c c c
43.
3 3 3
2 2
2
1000 10
10 10 10
10 10 100
x x
x x x
x x x
45.
36 2 3
22 2 2 2
2 4 2
64 1 4 1
4 1 4 4 1 1
4 1 16 4 1
t t
t t t
t t t
216
47.
6 3 6 3
32 3
22 2 2 2
2 4 2 2
2000 2 2 1000
2 10
2 10 10 10
2 10 100 10
y x y x
y x
y x y y x x
y x y y x x
49.
4 3
3 3
2 2
2
16 54 2 8 27
2 2 3
2 2 3 2 2 3 3
2 2 3 4 6 9
z z z z
z z
z z z z
z z z z
51.
312 4 3
24 4 4 2
4 8 4
125 5
5 5 5
5 5 25
p p
p p p
p p p
53.
222 1 1
36 625 5
1 16 6
5 5
y y
y y
55.
12 12
26 2
6 6
18 32 2 9 16
2 3 4
2 3 4 3 4
d d
d
d d
57. 2 2242 32 2 121 16v v
59.
2 2 2 24 16 4 4 4 2
4 2 2
x x x
x x
61.
22 225 49 5 7
5 7 5 7
q qq q
63.
2 2 22 36 2 6
2 6 2 6
t s t st s t s
65.
3 3 3
2 2
2
27 3
3 3 3
3 9 3
t t
t t t
t t t
67.
333
22
1 127 3
8 2
1 1 13 3 3
2 2 2
a a
a a a
69.
3 3 3 3
2 2
2
2 16 2 8 2 2
2 2 2 2
2 2 2 4
m m m
m m m
m m m
Section 4.7 Factoring Binomials
217
21 3 13 9
2 2 4a a a
71.
2 24 4 2 2
2 2 2 2
2 2
x y x y
x y x y
x y x y x y
73.
3 39 9 3 3
2 23 3 3 3 3 3
3 3 6 3 3 6
2 2 6 3 3 6
2 2 6 3 3 6
a b a b
a b a a b b
a b a a b b
a b a a b b a a b b
a b a ab b a a b b
75. 3 33
2 2
2
1 1 1 1
8 125 2 5
1 1 1 1 1 1
2 5 2 2 5 5
1 1 1 1 1
2 5 4 10 25
p p
p p p
p p p
77. 24 25w Prime
79. 2 22 21 1 1 1
25 4 5 2
1 1 1 1
5 2 5 2
x y x y
x y x y
81.
6 6
2 23 3
3 3 3 3
2 2 2 2
a b
a b
a b a b
a b a ab b a b a ab b
83.
26 2 3
3 3
3 3 3 3
2 2
64 8
8 8
2 2
2 4 2 2 4 2
y y
y y
y y
y y y y y y
85.
3 36 6 2 2
2 2 4 2 2 4
h k h k
h k h h k k
218
87.
36 2 3
22 2 2 2
2 4 2
8 125 2 5
2 5 2 2 5 5
2 5 4 10 25
x x
x x x
x x x
89. 2 2
2
2 3 2 3 2 3
4 9
x x x
x
91. 32 3
3
4 6 9 2 3 2 3
8 27
a a a a
a
93. 32 4 2 2 2 3
6 3
4 16 4 4
64
x y x x y y x y
x y
95. a. 2 2A x y 97.
2 2
1
x y x y x y x y x yx y x y
b. 2 2x y x y x y
c. 2 2
2 2 26 4 36 16 20 in
A x y
99.
3 3
2 2
2 2 1
x y x y
x y x xy y x y
x y x xy y
101. 5 2 3 2 2
2 3 2 3
2 2 3
2
576 9 64
9 (64 1) (64 1)
(9 )(64 1)
(3 )(3 )(4 1)(16 4 1)
a a a c c
a a c a
a c a
a c a c a a a
Problem Recognition Exercises:
1. A prime factor is an expression whose
only factors are 1 and itself.
3. When factoring binomials, look for:
Difference of squares: 2 2a b ;
Difference of cubes: 3 3a b ; or
Sums of cubes: 3 3a b .
5. Try factoring by grouping (2 terms and
two terms) or grouping 3 terms and one
term.
7. a. b.
Trinomial
2 2
2
6 21 45 3 2 7 15
3 2 10 3 15
x x x x
x x x
Problem Recognition Exercises: Factoring Summary
219
9. a. Difference of squares 11. a. Trinomial
b.
2 2
2 2
8 50 2 4 25
2 2 5
2 2 5 2 5
a a
a
a a
b.
2 2
2 2
14 11 2
14 7 4 2
7 2 2 2
2 7 2
u uv v
u uv uv vu u v v u vu v u v
13. a. Difference of cubes 15. a. Sum of cubes
b.
33 3 3
2
16 2 2 8 1 2 2 1
2 2 1 4 2 1
x x x
x x x
b.
33 3
2
27 125 3 5
3 5 9 15 25
y y
y y y
17. a. Sum of cubes 19. a. Difference of squares
b.
6 3 6 3
3 32
2 4 2 2
128 54 2 64 27
2 4 3
2 4 3 16 12 9
p q p q
p q
p q p p q q
b.
4 2 2
2 2
2
16 1 4 1
4 1 4 1
4 1 2 1 2 1
a a
a a
a a a
21. a. Grouping 23. a. Grouping
b.
22 2 212 36 6
6 6
p p c p cp c p c
b.
12 6 4 2
2 6 3 2
2 3 2 2
2 2 3
ax ay bx byax ay bx by
a x y b x yx y a b
25. a. Trinomial 27. a. Difference of squares
b.
2 25 14 3 5 15 3
5 3 3
y y y y yy y y
3 5 1y y
b. 2 2 2100 10
( 10)( 10)
t tt t
3 2 5 3 5
3 5 2 3
x x xx x
220
29. a. Sum of cubes 31. a. Trinomial
b. 3 3 3
2
27 3
( 3)( 3 9)
y y
y y y
b. 2 3 28 ( 7)( 4)d d d d
33. a. Perfect square trinomial 35. a. Grouping
b. 2 2 2
2
12 36 2( )(6) (6)
( 6)
x x x x
x
b. 22 5 2 5
(2 5) (2 5)
( )(2 5)
ax ax bx bax x b xax b x
37. a. Trinomial 39. a. Difference of squares
b. 210 3 4 (2 1)(5 4)y y y y b. 2 210 640 10( 64)
10( 8)( 8)
p pp p
41. a. Difference of cubes 43. a. Trinomial
b. 4 3
2
64 ( 64)
( 4)( 4 16)
z z z z
z z z z
b. 3 2 24 45 ( 4 45)
( 9)( 5)
b b b b b bb b b
45. a. Perfect square trinomial 47. a. Grouping
b. 2 2
2 2
2
9 24 16
(3 ) 2(3 )(4 ) (4 )
(3 4 )
w wx x
w w x x
w x
b. 2
2
60 20 30 10
10(6 2 3 )
10[2 (3 1) (3 1)]
10(2 )(3 1)
x x ax a
x x ax ax x a xx a x
49. a. Difference of squares 51. a. Difference of cubes
b. 4 2 2
2
16 ( 4)( 4)
( 2)( 2)( 4)
w w w
w w w
b. 6 2 3 3
2 4 2
8 ( ) 2
( 2)( 2 4)
t t
t t t
53. a. Trinomial 55. a. Perfect square trinomial
b. 28 22 5 (4 1)(2 5)p p p p b. 2
2 2
2
36 12 1
(6 ) 2(6 )(1) (1)
(6 1)
y y
y y
y
Problem Recognition Exercises: Factoring Summary
221
57. a. Sum of squares 59. a. Trinomial
b. 2 22 50 2( 25)x x b. 2 2 2 2
2 2
2
12 7 10
(12 7 10)
(4 5)(3 2)
r s rs s
s r r
s r r
61. a. Trinomial 63. a. Sum of cubes
b. 2 28 33 ( 3 )( 11 )x xy y x y x y b. 6 3 2 3 3
2 4 2 2
( )
( )( )
m n m n
m n m m n n
65. a. None of these 67. 2 2
2 2
2
( ) ( )
( )( )
( )( )( )
( ) ( )
x x y y x y
x y x yx y x y x y
x y x y
b. 2 4 ( 4)x x x x
69. 4 5 4
4
4
( 3) 6( 3) ( 3) (1 6( 3))
( 3) (1 6 18)
( 3) (6 19)
a a a a
a a
a a
71. 3 2
2
2
2
2
24(3 5) 30(3 5)
6(3 5) [4(3 5) 5]
6(3 5) [12 15]
6(3 5) 3(4 5)
18(3 5) (4 5)
x x
x x
x x
x x
x x
73. 2
2 2
2
1 1 1
100 35 49
1 1 1 12
10 10 7 7
1 1
10 7
x x
x x
x
75.
22 2
2
2
2 2
2 2
5 1 4 5 1 5
Let 5 1
4 5 5 1
5 1 5 5 1 1
5 6 5
x x
u x
u u u u
x x
x x
77. 4 4 2 2 2 2
2 2 2 2
2 2
16 (4 ) ( )
(4 )(4 )
(4 )(2 )(2 )
p q p q
p q p q
p q p q p q
79. 33 3
2
1 1
64 4
1 1 1
4 4 16
y y
y y y
222
81. 3 2 2 3
2 2
2 2
6 6
(6 ) (6 )
(6 )( )
(6 )( )( )
a a b ab b
a a b b a b
a b a ba b a b a b
83. 2
2 2
2
1 1 1
9 6 16
1 1 1 12
3 3 4 4
1 1
3 4
t t
t t
t
85. 2 2 2 212 36 ( 6)
( 6 )( 6 )
x x a x ax a x a
87. 2 2 2 22 81 ( ) 9
( 9)( 9)
p pq q p qp q p q
89. 2 2 2 2( 4 4) ( 2)
( ( 2))( ( 2))
( 2)( 2)
b x x b xb x b xb x b x
91. 2 2 2 2
2
4 2 4 2
4 ( )
(2 ( ))(2 ( ))
(2 )(2 )
u uv v u uv v
u vu v u v
u v u v
93. 6 2 3
6 2 3
2 (3 ) (3 )
(3 )(2 )
ax by bx ayax bx by ayx a b y a ba b x y
95. 6
3 2 2
3 3
2 2
2 2
64
( ) (8)
( 8)( 8)
( 2)( 2 4)( 2)( 2 4)
( 2)( 2)( 2 4)( 2 4)
u
u
u u
u u u u u u
u u u u u u
97. 8 4 2 2
4 4
4 2 2
4 2
1 ( ) 1
( 1)( 1)
( 1)( 1)( 1)
( 1)( 1)( 1)( 1)
x x
x x
x x x
x x x x
99. 2 2 ( )( ) ( )
( )( 1)
a b a b a b a b a ba b a b
101. 3 3 3 3 3 3 3 3
3 3
2 2
5 5 2 2 5 ( ) 2 ( )
( )(5 2 )
( )( )(5 2 )
wx wy zx zy w x y z x y
x y w z
x y x xy y w z
Section 4.8 Solving Equations by Using the Zero Product Rule
223
Section 4.8 Practice Exercises
1. a. quadratic e. ( ) 0f x ; yb. 0; 0 f. 1x ; 2x ; 2xc. Pythagorean; 2c g. lwd. quadratic h. 1
2bh
3. 210 3 10 3x x x x 5.
2 22 9 5 2 10 5
2 5 5
5 2 1
p p p p pp p pp p
7. 3 3 3 21 1 1 1t t t t t 9. The equation must be set equal to 0, and
the polynomial must be factored.
11. 2 3 0x x Correct form. 13. 23 7 4 0p p Incorrect form. The
polynomial is not factored.
15. 23 5a a Incorrect form. The equation is not set equal to 0.
17. a. 2 81 ( 9)( 9)w w w 19. a. 23 14 5 (3 1)( 5)x x x x
b.
2 81 0
( 9)( 9) 0
9 0 or w 9 0
9 or 9 9,9
ww w
ww w
b. 23 14 5 0
(3 1)( 5) 0
3 1 0 or 5 0
1 1 or 5 , 5
3 3
x xx x
x x
x x
21.
3 5 0
3 0 or 5 0
3 or 5 3, 5
x xx x
x x
23. 2 9 5 1 0
2 9 0 or 5 1 0
2 9 or 5 1
9 1 9 1 or ,
2 5 2 5
w ww w
w w
w w
224
25. 4 10 3 0
0 or 4 0 or 10 3 0
0 or 4 or 10 3
30 or 4 or
103
0, 4, 10
x x xx x xx x x
x x x
27.
0 5 0.4 2.1
5 0 or 0.4 0 or 2.1 0
no solution 0.4 or 2.1
0.4, 2.1
y yy y
y y
29.
2 6 27 0
9 3 0
9 0 or 3 0
9 or 3 9, 3
x xx x
x xx x
31.
2
2
2
2 5 3
2 5 3 0
2 6 3 0
2 3 3 0
3 2 1 0
3 0 or 2 1 0
3 or 2 1
1 13 or 3,
2 2
x x
x x
x x xx x x
x xx x
x x
x x
33.
2
2
10 15
10 15 0
5 2 3 0
5 0 or 2 3 0
0 or 2 3
3 30 or 0,
2 2
x x
x xx x
x xx x
x x
35. 6 2 3 1 8
6 12 3 3 8
3 15 8
3 23
23 23
3 3
y yy y
yy
y
37.
2
2
2
9 6
9 6
6 9 0
3 0
3 0
3 3
y y
y y
y y
yy
y
39.
2
2
2
9 15 6 0
3 3 5 2 0
3 3 6 2 0
3 3 2 2 0
3 2 3 1 0
p p
p p
p p p
p p pp p
3 0 or 2 0 or 3 1 0
2 or 3 1
p pp p
Section 4.8 Solving Equations by Using the Zero Product Rule
225
1no solution 2 or
31
2, 3
p p
41. 1 2 1 3 0
1 0 or 2 1 0 or 3 0
1 or 2 1 or 3
11 or or 3
21
1, , 32
x x xx x x
x x x
x x x
43.
2
2
3 4 8
12 8
20 0
5 4 0
5 0 or 4 0
5 or 4 5, 4
y y
y y
y yy y
y yy y
45.
2
2
2 1 1 6
2 3 1 6
2 3 5 0
2 5 1 0
2 5 0 or 1 0
2 5 or 1
5 5 or 1 , 1
2 2
a a
a a
a aa a
a aa a
a a
47.
22
2 2
2
2
7 169
14 49 169
2 14 120 0
2 7 60 0
2 12 5 0
2 0 or 12 0 or 5 0
12 or 5 12, 5
p p
p p p
p p
p p
p pp p
p p
49. 2 2
2 2 2
3 5 2 4 1
3 15 2 4 1
11 1
1 1
11 11
t t t t t
t t t t tt
t
51.
3 2
2
2 8 24 0
2 4 12 0
2 6 2 0
2 0 or 6 0 or 2 0
0 or 6 or 2 0, 6, 2
x x x
x x x
x x xx x xx x x
53.
3
3
2
16
16 0
16 0
4 4 0
w w
w w
w w
w w w
55.
3 2
2
2
0 2 5 18 45
0 2 5 9 2 5
0 2 5 9
0 2 5 3 3
x x x
x x x
x x
x x x
226
0 or 4 0 or 4 0
0 or 4 or 4
0, 4, 4
w w ww x x
2 5 0 or 3 0 or 3 0
2 5 or 3 or 3
5 or 3 or 3
25
, 3, 32
x x xx x x
x x x
57. Let x = the number
2
2
5 30
25 0
5 5 0
5 0 or 5 0
5 or 5
x
xx x
x xx x
59. Let x = the number
2
2
12
12 0
3 4 0
3 0 or 4 0
3 or 4
x x
x xx x
x xx x
61. Let x = the first consecutive integer
x + 1 = the second consecutive
integer
2
2
1 42
42
42 0
7 6 0
7 0 or 6 0
7 or 6
1 7 1 6 or 1 6 1 7
x x
x x
x xx x
x xx x
x x
The consecutive integers are –7 and –6
or 6 and 7.
63. Let x = the first consecutive odd integer
x + 2 = second consecutive odd
integer
2
2
2 63
2 63
2 63 0
9 7 0
9 0 or 7 0
9 or 7
2 9 2 7 or 2 7 2 9
x x
x x
x xx x
x xx x
x x
The consecutive odd integers are –9 and –
7 or 7 and 9.
65. Let x = the length
x – 2 = the width
67. Let x = the width
x + 5 = the length
Section 4.8 Solving Equations by Using the Zero Product Rule
227
2
2
2 35
2 35
2 35 0
5 7 0
5 0 or 7 0
5 or 7
or 2 7 2 5
x x
x x
x xx x
x xx x
x
The length is 7 ft and the width is 5 ft.
2
2
5 300
5 300
5 300 0
20 15 0
20 0 or 15 0
20 or 15
or 5 15 5 20
x x
x x
x xx x
x xx x
x
The width is 15 yd and the length is 20
yd.
69. a. Let b = the base of the triangle
b + 1 = the height of the
triangle
2
2
11 2 20
23 40
3 40
3 40 0
b b
b b
b b
b b
71. Let h = the height of the triangle
2h = the base of the triangle
2
2
12 25
2
25
25 0
5 5 0
5 0 or 5 0
5 or 5
2 2 5 10
h h
h
hh h
h hh h
h
The height is 5 ft and the base is 10 ft.
b. 8 5 0
8 0 or 5 0
8 or 5
1 5 1 6
b bb b
b bb
The base is 5 in and the height is
6 in.
215 6 15 in
2A
The area is 215 in .
73. Let x = the first positive consecutive
integer
x + 1 = second pos consecutive
integer
22
2 2
1 41
2 1 41
x x
x x x
22 2 40 0x x
75. a. Let x = the northern leg
x – 2 = the eastern leg
22 2
2 2
2
2 10
4 4 100
2 4 96 0
x xx x x
x x
22 2 48 0
2 6 8 0
x xx x
228
22 20 0
2 5 4 0
5 0 or 4 0
5 or 4
1 4 1 5
x x
x xx x
x xx
The consecutive positive integers are 4
and 5.
b.
6 0 or 8 0
6 or 8
2 8 2 6
x xx x
x
The alternative route is 8 mi + 6 mi
=14 mi.
10 10.25 hr
40 414 7
0.23 hr60 30
dtrdtr
The alternative route using
superhighways takes less time.
77. Let x = the first consecutive even
integer
x + 2 = second consecutive even
integer
x + 4 = third consecutive even
integer
2 22
2 2 2
2
2 4
4 4 8 16
4 12 0
2 6 0
2 0 or 6 0
2 or 6
2 6 2 8
4 6 4 10
x x x
x x x x x
x xx x
x xx x
xx
The lengths of the sides are 6 m, 8 m,
and 10 m.
79. Let r = the radius of the circle
2
2
2
2
2
2 0
2 0
0 or 2 0
0 or 2
r r
r r
r rr r
r rr r
The radius is 2 units.
81. a.
b.
2
2
3 0
3 0
0 or 3 0
0 or 3
0 0 3 0 0 0 0
f x x xx x
x xx x
f
83. a.
b.
2 6 7 0
7 1 0
7 0 or 1 0
7 or 1
f x x xx x
x xx x
20 0 6 0 7 0 0 7 7f
Section 4.8 Solving Equations by Using the Zero Product Rule
229
85.
12 1 2 0
21
0 or 2 0 or 1 0 or 2 02
2 or 1 or 0
10 0 2 0 1 2 0
21
2 1 0 02
f x x x x
x x x
x x x
f
x-intercepts: (2, 0), (–1, 0), (0, 0)
y-intercept: (0, 0)
87.
2
2
2
2 1 0
1 0
1 0
1
0 0 2 0 1 0 0 1 1
f x x x
xx
x
f
x-intercepts: (1, 0)
y-intercept: (0, 1)
89. 3 3 0
3 0 or 3 0
3 or 3
g x x xx x
x x
x-intercepts: (–3, 0), (3,0)
Graph d.
91. 4 1 0
4 0 or 1 0
1
f x xx
x
x-intercepts: (–1, 0)
Graph a.
93. a. The function is in the form
2s t at bt c
d. At 0 sec and 100 sec, the rocket is at
ground level (height = 0).
2
2
2
4.9 490 485.1
4.9 490 485.1 0
4.9 100 99 0
4.9 1 99 0
4.9 0 or 1 0 or 99 0
1 or 99
s t t t
t t
t t
t tt t
t t
The height is 485.1 m at 1 sec and 99 sec.
b.
24.9 490 0
4.9 100 0
4.9 0 or 100 0
0 or 100
s t t tt t
t tt t
c. t-intercepts (0, 0), (100, 0)
95.
2 7 10 0
5 2 0
5 0 or 2 0
5 or 2
f x x xf x x x
x xx x
x = 5 and x = 2 represent the
x-intercepts.
97.
2
2
2 1 0
1 0
1 0
1
f x x x
f x xx
x
x = –1represents the x-intercept.
230
99.
2
2
6 5 0
( 6 5) 0
1 ( 5) 0
1 0 or 5 0
1 or 5
f x x x
f x x xf x x x
x xx x
1x and 5x represent the
x-intercepts.
101.
2
2
2
2
2 2
2 2 7 156
7 78
7 78 0
13 6 0
13 0 or 6 0
13 or 6
SA r rh
r r
r r
r rr r
r rr r
The radius is 6 ft.
103. Let l = the length w = the width
2 2 28
2 28 2
14
l ww lw l
2
14 48
14 48
A l l
l l
20 14 48
0 8 6
l ll l
8 0 or 6 0
8 or 6
14 8 6
l ll l
w
The length is 8 ft and the width is 6 ft.
105.
2
2 and 2
2 2 0
4 0
x xx x
x
107.
2
0 and 3
0 3 0
3 0
x xx x
x x
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