Chapter 5
EXPONENTS AND POLYNOMIALS
5.1 Laws of Exponents
Exercises
2. The quotient rule of exponents states that
8.
2
77749
88864
⎛⎞ ⎛⎞⎛⎞
==
⎜⎟ ⎜⎟⎜⎟
⎝⎠ ⎝⎠⎝⎠
10.
() ()()
2
0.2 0.2 0.2 0.04==
12.
() ()()
2
0.3 0.3 0.3 0.09−=− =−
14.
() ()()()
3
444464−=−−−=−
16.
() ()()()()
4
666661296− =−−−−=
34. 45
ab⋅
Cannot be simplified
36. 0101
, or yy y y y
+
⋅= =
38. 43
ab
Cannot be simplified
12 1
x
46.
7
75 2
5
xxx
x
−
==
48.
5
a
b
Cannot be simplified
50. 40 40 4
rr r r
−
÷= =
52. 2112 4
ttt t t
++
⋅⋅ = =
226 226 38
Chapter 5 Exponents and Polynomials
88
66.
()
11
55
yy
−
−=−
78. 3
3
x
xy y
−=
80. 1r
rs s
−=
82.
2
24
4
5
5a
ab b
−
−=−
84. 33 6
6
1
tt t t
−− −
⋅= =
86. 42 2
ss s
−
⋅=
98.
1
5
6
yy
y
−
−=
110.
2
32
3
b
ab a
−⋅=
112.
53
35
mn
nm
−
−=
114. a.
()
()
1
28,000 1.25
28,000 1.25
28,000
1.25
22, 400
t−
−
=
=
$22,400
b.
()
28,000 1.25
t−
Section 5.2 More Laws of Exponents and Scientific Notation
89
Mindstretchers
1. a. The base was incorrectly changed from
positive to negative.
2. It depends on the value of .x 2
x is larger
than 2
x− when 1x<− or 1;x> 2
x− is
larger than 2
x when 1 0x−< < or
01.x<< The values are equal when
5.2 More Laws of Exponents
and Scientific Notation
Exercises
2. The power rule of exponents states that to
raise a power to a power, multiply the
exponents and leave the base the same.
12.
()
3
3339
2 2 2 512
⋅
===
14.
()
3
55315
0000
⋅
===
18.
()
ppp
==
20.
()
3
3339
nnn
⋅
==
6
55630
1
−−⋅−
30.
(
)
()
(
)
3
33
339
5 5 125xxx−=− =−
32.
() () ( )
3
3333
23 23 227 54tttt−=−=−=−
34.
() ()
3
333 3
111
55 125
5
ttt
t
−
−=−=−=−
Chapter 5 Exponents and Polynomials
90
48.
()
6
2
23 2 2232
4
4
22 y
xy x y x
−−⋅⋅
==
56.
3
66318
3
464
4
yyy
⋅
⎛⎞
==
⎜⎟
⎝⎠
58.
()
4
2248
4
33412
1
xxx
yyy
⋅
⋅
⎛⎞
−=− =
⎜⎟
⎝⎠
60.
11
33
3
b
bb
−
⎛⎞ ⎛⎞
==
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
62.
2
2224
525210
339
nnn
www
⋅
⋅
⎛⎞
==
⎜⎟
⎝⎠
64.
()
2
23 222
2
st st s t
st
⎛⎞
==
⎜⎟
⎝⎠
74.
22
542428
455210
5525
5
qppp
pqqq
−⋅
⋅
⎛⎞⎛⎞
===
⎜⎟⎜⎟
⎜⎟⎜⎟
⎝⎠⎝⎠
75,500, 000,000
=
82. 32.1
2.1 10 1000
0.0021
−
×=
=
84. 8
100, 000, 000 1 10=×
86. 4
0.00017 1.7 10−
=×
88. 11
154,800,000,000 1.548 10=×
90. 8
0.00000005672 5.672 10−
=×
92.
11
8
Standard Scientific Notation Scientific Notation
Notation (written) (on a calculator)
975,000,000,000 9.75 10 9.75E11
500,000,000 5 10 5E8
×
×
Section 5.2 More Laws of Exponents and Scientific Notation
91
98.
()( )
912
8.6 10 4.4 10
−
××
100.
()()
43
4
3
43
1
3.0 10 1 10
310
110
310
310
−
×÷×
=×
=×
=×
104.
(
)
(
)
67
6
7
67
1
8.4 10 4.2 10
8.4 10
4.2 10
210
210
−
−
×÷×
=×
=×
=×
⎝⎠
110.
()
3
737321
3
73
3339
228
2rrr
rs sss
⋅
−
⋅
⎛⎞
===
⎜⎟
⎝⎠
5
22510
5
mmm
⋅
⎛⎞
116.
()( )
32
4.1 10 2.7 10
−−
××
118. 8
2.5 10 m 0.000000025 m
−
×=
7
2 10 m 0.0000002 m
−
×=
120.
()
2
2
2
4
A
r
r
π
π
=
=
2
8
1.4 0.00000001 1.4 10 cm
−
×=×
130. 13
610× operations per second
132. a. 19
2.689 10 , or 2.689E19 molecules per
×
13
1.264410562 10 m
=×
Mindstretchers
1. Answers may vary.
Chapter 5 Exponents and Polynomials
92
5.3 Basic Concepts
of Polynomials
Exercises
8. Polynomial terms are usually written in
descending order of degree.
10. Polynomial
16. Not a polynomial
18. a. Terms: 3
6,4,and 3yy−−
b. Coefficients: 6, 4, and 3−−
26. 43 2
3 2 25; degree 4yy yy−++ −+
28. 53
3 5 8 3; degree 5xxx+++
34.
33
22
44
Constant Leading Leading
Polynomial Term Term Coefficient
58 8 5 5
10 10 1
234 4 2 2
599 1
xx
xx
xx x
xx x
+
−+ − −
−+
−+− −
40. 323
2
57 1
761
xxx xx
xx
+− −−+
=− + −
42. 23 2
882039
yy y yy
+−+++
46. 32 0
0 ,0 , and 0 , or 0xx x
48.
()
5115011
011
11
a
+= +
=+
=
() ()
() ( )
2
2
32131211
31 2 1 1
321
2
yy
++=−+−+
=+−+
=−+
=
Section 5.3 Basic Concepts of Polynomials
93
52.
() ()
3
3
0.1 4.1 9.1
0.1 3.14 4.1 3.14 9.1
xx+−
=+−
54. 3
68; degree 3xx+
56. 2
3
23;Binomial
8 ; Monomial
pp
x
−
−
60. 34 24 2
432
483654
3423
nn nnn
nnn
−+++−
=− + + +
68.
() ()
2
2
0.58 4.96 41.48
0.58 8 4.96 8 41.48
xx++
=++
70.
() () ()
32
32
88.167 184.500 114.333 65,337
88.167 1 184.5 1 114.333 1 65,337
xxx−+ −+
=− + − +
Bi- Two Answers may vary.
Tri- Three Answers may vary.
Poly- Many; several Answers may vary.
2. a. For example, for 0, 8, 21, and 36,n= the
value of the polynomial is 41, 113, 503, and
Rectangle 2
lw
3
Degree of the
Geometric Figure Polynomial Polynomial
Cube 3
Rectangular solid 3
e
lwh
of the polynomial and the exponent to which
the unit of measure for volume is raised are the
same (3).
Chapter 5 Exponents and Polynomials
94
5.4 Addition and
Subtraction of
Polynomials
Exercises
6.
(
)
(
)
22
22
2
381034
381034
52
xx xx
xx xx
x
+−+ −+
=+−+−+
=+
10.
()()
()
32 23 2
223
24 3 6
223
xxyxyy xy
xy xy y
−+++ −
+−−+
12. 2
2
45
10
315
tt
t
tt
++
−+
++
18.
(
)
(
)
32 32
32 32
32
810 27 3
810 2 7 3
617 2
xxxxxx
xxxxxx
xxx
−+−++
=− +−−−
=− −
20.
(
)
(
)
22
832571
tt tt
−+− −−
444 224
4224
83 7
77 4
pqp pqq
ppqq
=−−+ −
=+ −
26.
2
3102
xx
+−
2
10 7
813
x
xx
−−
−−
33 2
32
332
13 7 10
52
13 12 8
xy x y
xy
xy x y
+−
+
+−
332
12 13 8xxyy+−
Section 5.4 Addition and Subtraction of Polynomials
95
40.
()()
2
3
2
3
2
3
42 81
42 81
41
nn nn
nn nn
nn n
−+− −+
=−+−+−
=−++
42.
()
()
()
2
2
2
2
2
5912
5912
6101
yy y y
yyy y
yy
++ −+ +
= ++−++
=++
48.
()
(
)
2
2
2
34 5
34 5
84
xxx
xxx
xx
−− −
=−−+
=− + −
50.
(
)
34 34abc abc−−=−+
60.
()
32
32
32
32
32
327 1651 3 178,279
149 745 310 34,022
327 1651 3 178, 279
149 745 310 34,022
178 906 307 144,257
xxx
xxx
xxx
xxx
xxx
−++
−− ++
−++
−+−−
−−+
(
)
32
178 906 307 144,257xxx−−+ thousand
Mindstretchers
1. a. 76 123 199+=
b. Third number: ab+;
fourth number:
()
2.ab ba b++=+
Chapter 5 Exponents and Polynomials
96
5.5 Multiplication of
Polynomials
Exercises
2.
()()( )
()
11
2
32 32
6
xx x
x
+
−=−⋅
=−
10.
()( )
()
()
2 47 1427
59
86 86
48
st s t s t
st
++
=⋅
=
12.
()()
22
2
2
10 10
100
pp
p
−=−
=
14.
33
333
9
11
33
1
27
nn
n
⋅
⎛⎞⎛⎞
=
⎜⎟⎜⎟
⎝⎠⎝⎠
=
22.
()
()
()()
()
()
2
2
23
54
54 4
20 4
xx x
xx x x
xx
−
=+−
=−
24.
()
()
()
3
3
24
28
28 2
16 2
yyy
yy y y
yy
−
=+−
=−
30.
()
()
() ()
2
2
32
43 2
43 4 42
1248
xx x
xx x x x
xxx
−+
=+−+
=−+
32.
()
()
()
()()()()()
2
2
32
10 1 2
10 2 2 1 2
20 2 2
xx x
xxxx x
xxx
+−
=++−
=+−
34.
(
)
(
)
() ()
32 4
34 24
69103
363
xxx x
xx xx
+−+−
=− + −
Section 5.5 Multiplication of Polynomials
97
42.
()
2
2
7359
715 27
15 34
xxx
xx x
xx
−+ −
=− + −
=−
48.
() ( )
32222
433 42 23
42 4 33 23
25 7 910
10 2 63 70
63 10 2 70
st s t st s t
st st st st
st st st st
−− −
=−− +
=− + − +
52.
(
)
(
)
()()
()()
()( )
()( )
2
14
F:
O: 4 4
I: 1
L: 1 4 4
xx
xx x
xx
xx
++
=
=
=
=
2
2
44
54
xxx
xx
=+++
=++
56.
(
)
(
)
()()
2
33
F:
xx
xx x
+−
=
58.
(
)
(
)
()()
()()
()()
2
85 4
F: 8 8
O: 8 4 32
I: 5 5
xx
xx x
xx
xx
++
=
=
=
()()
()( )
I: 1 3 3
L: 1 2 2
uu
−=−
−− =
2
2
12 8 3 2
uu u
=−−+
()( )
()( )
I: 1 7 7
L: 1 3 3
xx
=
−=−
2
2
49 21 7 3
49 14 3
xxx
xx
=−+−
=−−
64.
(
)
(
)
()()
()( )
2
F:
O:
xyxy
xx x
xy xy
+−
=
−=−
()( )
()()
()()
2
O:
I: 4 4
L: 4 4
xy xy
yx yx
yy y
−=−
=
−=−
Chapter 5 Exponents and Polynomials
98
68.
(
)
(
)
()()
2
54
F: 5 5
xyxy
xx x
+−
=
70.
(
)
(
)
()( )
()( )
()( )
()( )
76 1
F: 6 6
O: 1
I: 7 6 42
L: 7 1 7
xy
xy xy
xx
yy
+−
=
−=−
=
−=−
6427xy x y=−+−
72. 2
(
(
44
2
aa
a
−+
+
74. 2
2
(
(
291
83
6273
nn
n
nn
−−
+
−−
78.
()
()()
()
()
2
23
83 8
838
yyy
yy y
−+
=− +
80.
(
)
(
)
(
)
()
()
2
23
23
aa ba b
aabab
−+ −
=− − −
32 2 2
32 2
326
6
aababab
aabab
=− + − +
=− + +
82.
()
22
232
32
85321
815105
15 2 5
tttt
tttt
ttt
+−+
=+ − +
=−+
84.
()
223
38 24 3ppp p p−= −
()()
L: 5 7 35
−−=
2
2
7535
12 35
uuu
uu
=−−+
=− +
90.
(
)
(
)
27 5 312
39 27hk hkl hk l−=−
92.
(
)
(
)
()( )
2
68 3
F: 8 8
aa
aa a
+−
=
Section 5.5 Multiplication of Polynomials
99
94.
()
2
6252512
62510 24
Add
Add
−= −
−= −
96.
1100
Rpx
pp
=
⎛⎞
=− +
⎜⎟
98. a.
()
3
3
4
3
45
3
r
r
π
π
=+
b.
()
()()()
3
3
4
3
45
3
4555
3
r
r
rrr
π
π
π
=−
=−−−
98. c. 32
4 500
20 100
33
rr r
π
ππ π
+++
2
33
1000
40 3
r
ππ
+
1. The sum of the areas of the small rectangles is
equal to the total area of the outer rectangle,
whose sides are 9x+ and. 2.x+ The total area
of this outer rectangle is represented by the
product
(
)
(
)
92.xx++ Therefore, the sum of
()()()
2
22
22
2
xy xyxy
xxyxyy
xxyy
+=+ +
= +++
=+ +
()()()
()
()
32
22
32 2 2 23
2
22
xy xy xy
xxyyxy
xxyxyxyxyy
+=+ +
=++ +
=+ + + + +
Chapter 5 Exponents and Polynomials
100
5.6 Special Products
Exercises
2. In the formula for the square of the sum of
two terms, the middle term of the trinomial
is positive.
8.
() ()()
222
2
8288
16 64
nnn
nn
+=+ +
=+ +
10.
( ) ()()()
22
2
2
10 2 10 10
20 100
bbb
bb
−=− +
=− +
18.
( ) () ()()()
22 2
2
21 2 2211
441
xxx
xx
−= − +
=−+
20.
( )() ()()()
22 2
2
11 3 11 2 11 3 3
121 66 9
mmm
mm
+= + +
=++
28.
( ) () ()( ) ( )
222
22
2222
44
pq p pq q
ppqq
−+ = − +
=− +
30.
(
)
(
)
(
)
(
)
(
)
222
22 2 22 2
4224
5525
25 10
ac a ac c
aacc
−= − +
=− +
32.
22
88 8
rr r
+−=−
() ()
2
91
81 1
x
x
=−
=−
38.
2
2
2
11 1
44 4
1
mm m
m
⎛⎞⎛⎞ ⎛⎞
−+=−
⎜⎟⎜⎟ ⎜⎟
⎝⎠⎝⎠ ⎝⎠
=−
44.
()()()()
22
10 3 10 3 10 3
100 9
ts ts t s
ts
+−=−
=−
46.
(
)
(
)
(
)
(
)
() ( )
22
2
2 332 32 32
32
94
ssss
s
s
+−=+ −
=−
=−
Section 5.6 Special Products
101
54.
()()
()
()()
22
2222
44
xyxyx y
xyxy
xy
+− +
=− +
=−
62.
( ) () ()()()
22 2
22
35 3 235 5
93025
xy x xy y
xxyy
−= − +
=− +
66. a.
The longest possible true dimensions of
the wood are xe+ by .xe+ The
shortest possible true dimensions of the
wood are
xe− by .xe−
70. b. 3% 0.03r==
() ()
() ( )
2
1000 2000 0.03 1000 0.03
1000 2000 0.03 1000 0.0009
1000 60 0.9
A=− +
=− +
=−+
b.
2
2
30 29 30 30
10 10 10 10
1
30 10
1
×=+ −
⎜⎟⎜⎟
⎝⎠⎝⎠
⎛⎞
=−
⎜⎟
⎝⎠
(
)
2
n from the larger square
(
)
221nn++ results
in the expression 2 1.
n+ Since 2n is even, it
follows that 2 1
n+ is odd.
3. The four rectangles make up a square that
Chapter 5 Exponents and Polynomials
102
5.7 Division of Polynomials
Exercises
2. To divide a polynomial by a monomial,
divide each term in the polynomial by the
monomial and then add.
8.
22
21
35 35
77
5
5
yy
yy
y
y
−
−−
=⋅
=−
=−
14.
54 54
22
52 41
33
44
2
2
2
2
xy xy
xy xy
xy
xy
−−
−−
=⋅
−
−
=⋅
=
18.
32 32
22
32 11 21
10 10
8
8
5
4
5
4
xyz xyz
xyz xyz
xyz
xz
−−−
−−
=⋅
=−
=−
43 4 3
12 15 12 15
mm m m
+=+
85
x
=− −
26.
32 3 2
222
64 6 4
222
32
aa a a
aaa
a
−=−
=−
34.
864
4
86 4
444
10 4 6
2
10 4 6
ab ab ab
ab
ab ab ab
−+
−
=−+
Section 5.7 Division of Polynomials
103
40. 2
2
53
6 1 30 13 3
30 5
18 3
18 3
0
x
xxx
xx
x
x
+
−+−
−
−
−
44. 2
2
25
7 2 19 35
214
535
535
0
x
xxx
xx
x
x
+
+++
+
+
+
52. 2
2
313 8
3
8
x
xxx
xx
−−−
−
−
8
xx
−
+−
5
5
0
x
x
−
−
56.
2
32
32
2
212 48
2
48
x
xxxx
xx
x
−
++−−
+
−−
Chapter 5 Exponents and Polynomials
104
62.
2
32
2
3
2
2
1
1001
0
1
1
xx
xxx x
xx
xx
xx
x
x
++
−++−
−
+
−
−
−
68. 2
2
92
4136 2
36 9
82
82
0
x
xxx
xx
x
x
−
++−
+
−−
−−
2
216
324
324
0
xx
x
x
−
−
−
76. a.
()
0.01 0.49 dollarsx+
b. 0.01 0.49 dollars
x
+
55
0
r
−+
Long division yields 2
555,rr++ which is equal
to the sum of the first three terms.
80.
2
32
350600
3 300 9 750 16,800 180,000
xx
xxx x
−+ +
+−− + +
Section 5.7 Division of Polynomials
105
Mindstretchers
1. The trinomial is a factor of the polynomial
if the remainder resulting from the division
is 0.
2. a.
2
Divisor Dividend Quotient
3
11 2
xxx x
+−+ −+
3. 2
2
23
12 5 3
22
y
yyy
yy
+
+++
+