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179
Chapter 7 Polynomials
Section 7.1 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
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
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 7.2 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
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
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
189
c.
(50, 22.6) At t = 2 sec, the position of the rocket is (50, 22.6).
Section 7.3 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
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 .
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
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 7.4 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
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
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
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 .
203
87. a. 1 8 13 5
8 5
8 5 0
Quotient: 8 5x
b. YesYes
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