Chapter 5
2. The quotient rule of exponents states that
when powers of the same base are divided,
() ()()()
32. 8189
Cannot be simplified
56.
(
)
(
)
aab bc aa b bc
=⋅ ⋅ ⋅⋅
()
nn
33n
98.
6
y
−=
()
28,000
()
()
0
28,000
=
=
$28,000; this value represents the
118. a. 5
6 , or 7776
89
b. The exponents were multiplied, instead of
c. The bases were multiplied.
d. The bases were divided, and the
than 2
x− when 1x<− or 1;x> 2
x− is
larger than 2
x when 1 0x−< < or
1x=− or 1.x= When x is 0, 2
x− is
undefined.
By the product rule, this is equal to 9
10 ,
and Scientific Notation
Exercises
6. A number is written in scientific notation
integer and a is greater than or equal to 1
but less than 10.
8. To convert a number from standard to
scientific notation, move the decimal point
()
()
()
()
12
2
()
()
()
30
y
()
()()
(
)
(
)
() () ( )
()
()
()
()
()
6
()
68
Chapter 5 Exponents and Polynomials
90
66.
===
⎜⎟
⎜⎟
4
82
0.000833
=
7.55 10 7.55 10,000,000, 000
×=×
0.0021
=
88. 11
154,800, 000,000 1.548 10=×
92.
Notation (written) (on a calculator)
0.0000004988 4.988 10 4.988E 7
()
()
43
+
()()
()
5.733 10
=×
91
()( )
()
()()
4.2 10
()
33515
⋅
⎜⎟
⎝⎠
()
3339
⋅
⎜⎟
⎝⎠
()
()
()
343
12
23
54
x
x
x
⋅
=−
=−
()( )
()
()( )
13
1.264410562 10 m
=×
Mindstretchers
()
m
a
⋅
mn mn
⋅
() ()
multiplicative inverses.
3. Answers may vary.
92
of Polynomials
18. a. Terms: 3
6,4,and 3yy−−
20. 2
3; Monomial
x
24. 57; degree 1x−+
32. 2
5 ; degree 2x
10 10 1
xx
+
−+ − −
011
10 11
=+
=+
() ()
() ()
() ()
() ( )
34 22 1
321
=++
=
=−+
93
3
8 ; Monomial
x
−
() ()
() ()
() ()
() ( )
59 83 9
45 24 9
54 8 2 9
=−−
=−−
=−−−
60. 34 2 4 2
62. 23 2
64. A polynomial in x; degree 3
() ()
0.5 400 0.5 20
=−
118.28 118
$118 million
2. a. For example, for 0, 8, 21, and 36,n= the
value of the polynomial is 41, 113, 503, and
3. a.
Triangle 0.5 2
Trapezoid 0.5 0.5 2
bh
hb hB
+
Degree of the
Subtraction of
Polynomials
2.
(
)
(
)
2
4.
(
)
(
)
92 3
yy yy
++−−
6.
(
)
(
)
()
()
42
410
19 26 25
pq q p
ppqq
+− + −
=++
()()
()
223
32 23 2
32 23
223
24 3 6
22 3 3
xy xy y
xxyxyyxy
xxyxyy
+−−+
=− +++ −
=− + −−
12. 2
45
tt
++
(
)
(
)
32
617 2
xxx
=− −
20.
(
)
(
)
22
832571
tt tt
−+− −−
(
)
()
()( )
()
2
3102
52
xx
xx
+−
−+ −
28.
2
26
813
xx
xx
+−
−−
()
33 2
33 2
32
13 7 10
52
xy x y
xy
+−
+
32.
()
89
xxy
−+
95
()
()
()()
2
23
35 6
pp p p
=+−−−+
()()
2
3
42 81
nn nn
=−+−+−
()
()
()
()()( )
()
()
()
22
3217
mn mn
=−+
()
(
)
2
34 5
xxx
=−−+
()
()
()( )
56. 32
32
72 7
ttt
−− ++
() ()()
324 21
303
=−
=
()
32
149 745 310 34,022
xxx
−− ++
()
Mindstretchers
()
2.ab ba b++=+
(
)
(
)
813 1321 2134.ab a b a b++ + =+
()
235 2
xxxx
−+− ++
()()
but
()( )
22
43 2 37xx xx+−− −+
2710.xx=− + −
96
Polynomials
Exercises
()()( )
()
2
6
x
=−
7
42
y
=−
()()
()
()
()
()
()
()
100
100
a
=−
()( )
()
59
48
st
=
100
p
=
33
333
11
⋅
⎛⎞⎛⎞
()()
()
()
33
7
82 1
nnn
=−⋅ ⋅− ⋅ ⋅
()
()
()
()( )
()
32
35
417
28
mmm n n
mn
=⋅−⋅ ⋅⋅ ⋅
=−
37
yy
=−
()()
()
()
23
20 4
xx
=−
()
()
3
24
28 2
16 2
yy y y
yy
=+−
=−
()
() ()
77
87
59 5 3
pp p
=− − −
()
432
8
ttt
=+ +
()
() ()
2
32
43 4 42
1248
xx x x x
xxx
=+−+
=−+
()
()
()
()()()()()
2
32
10 2 2 1 2
20 2 2
xxxx x
xxx
=++−
=+−
() ()
76 5 4
318 27 30
xx xx
=− − + −
()
38.
()()()()
22 5
35 7
3
33
mnmn
mn mn
+
=+
() ( )
45 56
28 16
xy xy
=− +
Section 5.5 Multiplication of Polynomials
97
()
2
2
715 27
15 34
xx x
xx
=− + −
=−
(
)
32 2
16 14 4 6
xxxx
=+−−
32
19 11 30
xx x
=− + −
() ( )
433 42 23
()
()( )
()
33
15 68 7
ppqq
=− −
()()
()()
()( )
()( )
2
F:
O: 4 4
I: 1
L: 1 4 4
xx x
xx
xx
=
=
=
=
2
54
xx
=++
()()
()( )
()()
()()
2
F:
O: 2 2
I: 4 4
L: 4 2 8
nn n
nn
nn
=
−=−
−=−
−−=
2
68
nn
=−+
56.
(
)
(
)
()()
()( )
()()
()( )
33
F:
O: 3 3
I: 3 3
L: 3 3 9
xx
xx x
xx
xx
+−
=
−=−
=
−=−
2
2
339
9
xxx
x
=−+−
=−
(
)
(
)
()()
()()
()()
()()
2
F: 8 8
O: 8 4 32
I: 5 5
xx x
xx
xx
=
=
=
()()
()( )
()( )
2
O: 4 2 8
L: 1 2 2
uu u
uu
−=−
−− =
()()
()()
()( )
O: 7 3 21
xx
−=−
2
2
49 21 7 3
49 14 3
xxx
xx
=−+−
=−−
()()
()( )
()()
()( )
O:
I:
xy xy
yx yx
−=−
=
22
22
xxyyxy
xy
=−+−
=−
()()
()( )
()()
()()
O:
I: 4 4
xy xy
yx yx
−=−
=
22
22
44
34
xxy yxy
xxyy
=−+ −
=+ −
Chapter 5 Exponents and Polynomials
98
68.
(
)
(
)
()()
()( )
()()
()()
2
54
O: 5 5
I: 4 4
L: 4 4
xyxy
xy xy
yx yx
yy y
+−
−=−
=
−=−
22
22
5544
54
xxyyxy
xxy y
=−+−
=−−
()( )
()( )
()( )
I: 7 6 42
L: 7 1 7
yy
=
−=−
6427xy x y=−+−
32
44
248
a
aaa
aaa
+
−+
−−+
74. 2
291
83
nn
n
−−
+
32 2
223
yx
xxyxy
xy xy y
−
−− −
++
()
()()
()
()
(
)
()
()
()
()
()
()
()
22
23
33
34
F: 8 8 64
O: 8 8
I: 3 8 24
L: 3 3
yy
yy y
yy
yy y
=
=
−=−
−=−
(
)
(
)
(
)
()
()
23
23
aa ba b
aabab
−+ −
=− − −
(
)
()
()
()
()()
()()
23
22
2
2
F:
O: 3 3
I: 2 2
L: 2 3 6
aa a
abab
ab a a b
ab b ab
−=−
−−=
−=−
−−=
32 2 2
326
aababab
=− + − +
()
232
32
815105
15 2 5
tttt
ttt
=+ − +
=−+
()()
()( )
()()
()()
O: 7 7
I: 5 5
L: 5 7 35
uu
uu
−=−
−=−
−−=
2
7535
uuu
=−−+
()()
()()
()( )
L: 3 3
ww
−−=
()( )
()( )
()( )
()( )
2
F: 8 8
O: 3 3
I: 6 8 48
L: 6 3 18
aa a
aa
aa
=
−=−
=
−=−
2
2
8 3 48 18
aaa
=−+−
99
2
2
62510 24
610 2425
Add
Ad d
−= −
=−+
96.
1100
pp
⎛⎞
=− +
()
()()()
()
()
2
3
410 25 5
3
rr r
π
=+++
()
3
33
()
()()()
()
()
3
2
4
3
4555
3
410 25 5
3
rrr
rr r
π
π
=−−−
=−+−
()
32
32
415 75 125
3
4 500
20 100
33
rr r
rr r
π
ππ π π
=−+−
=− + −
98. c. 32
32
20 100
33
4 500
20 100
rr r
rr r
ππ π
π
ππ π
+++
⎛⎞
−−+−
⎜⎟
2
1000
40 3
r
ππ
+
product
(
)
(
)
92.xx++ Therefore, the sum of
211 18,xx=+ + is equal to the total area of the
outer rectangle,
(
)
(
)
92.xx++
()
()()()
22
xxyxyy
= +++
()()()
()
()
22
32 2 2 23
32 23
2
22
33
xy xy xy
xxyyxy
xxyxyxyxyy
xxyxyy
+=+ +
=++ +
=+ + + + +
=+ + +
The first and last number in each row is 1 and
every other number in each row is the sum of
the two numbers directly above it.
3. a. 53 89 4717⋅=
100
second term.
() ()()
() ()()
2
16 64
nn
=+ +
( ) ()()()
2
20 100
bb
=− +
( ) () ()()()
2
( ) () ()()()
2
441
xx
=−+
( )() ()()()
2
121 66 9
mm
=++
2
111
⎛⎞ ⎛⎞⎛⎞
( ) () ()()()
22
96
sstt
=++
( ) () ()()()
22
92416
mmnn
=− +
( ) () ()( ) ( )
(
)
(
)
(
)
(
)
(
)
()()()()
2
49 4
y
=−
() ()
2
2
11 1
1
16
m
⎛⎞⎛⎞ ⎛⎞
=−
()()()()
()()()()
46.
(
)
(
)
(
)
(
)
2
2 332 32 32
94
ssss
s
+−=+ −
=−
()()
()
()
()
543
52.
(
)
(
)
(
)
(
)
22
3535 3 5
xyxy x y
+−=−
Section 5.6 Special Products
101
56.
()()()()
() ()()
() ( )
2
()
b.
()
12 6 12 3 6
n+= +
()()
()
()()
()
12121 2
2222
1212
44
12
aT T T T
aT T
=− +
=−
()
2. No, it is not possible that the difference is an
even number. Let n and 1n+ represent two
3. The four rectangles make up a square that
(
)
,aa b− and the area of the remaining green
rectangle is
(
)
.ba b− The combined area is
102
5.7 Division of Polynomials
4. To check if the quotient of two polynomials
21
−
10.
22
=⋅
2
3
a
=
52 41
2
xy
−−
=⋅
16.
22
23
10
xy
=⋅
=−
22
8
8
xyz xyz
333
mmm
26.
222
2222
32.
222
777
xy xy xy
34.
4
36.
23
6
xy
103
10
x
xxx
xx
−
++−
+
2
30 5
xx
−
2
10 4
xx
+
44. 2
7 2 19 35
xxx
+++
46. 2
232 3
xxx
++−
48. 2
2
227
2
47
48
xxx
xx
x
x
−++
−
+
−
50. 2
2
15 2 3
xxx
−−−
52. 2
2
313 8
3
xxx
xx
−−−
−
54.
32
57115
5
xxx x
xx
−−+−
−
56.
212 48
6
xxxx
++−−
−
32
32 96
96
x
x
−
−−
104
60.
2
3
33
xx
+
+
62.
66.
3
843
3
15 21 24
ab
ab ab ab
68. 2
2
36 9
xx
+
70.
222
ttt
2
412
xx
−
74. 2
8 2 13 24
324
xxx
x
−−−
−
x
+
2
3
55
rr
−+
Long division yields 2
555,rr++ which is equal
80.
2
3 300 9 750 16,800 180, 000
+−− + +
()
Section 5.7 Division of Polynomials
105
Mindstretchers
2. a.
2
3
11 2
1
xxx x
x
+−+ −+
+
b.
2345 1
xxxx x
−+−++
+
2
23
12 5 3
22
33
33
0
y
yyy
yy
y
y
+
+++
+
+
+
23 1
22
1
yy
y
y
+> +
>−
>−
The quotient 23y+is larger in value than
the divisor for all values y greater than 2.−
