Section 5.6 Long Division of Polynomials; Synthetic Division
70. First, compute the difference:
()( )
32 32
32
42 12 25
2334
xxx xxx
xxx
+−−− −+−
=+−+
Now, complete the division:
32
2334
2
xxx
x
+−+
+
71.
A
lw=⋅
so
32
353
1
Ax x x
lwx
+++
== +
2
32
23
1353
xx
xx x x
++
++++
72.
A
bh=⋅
so
32
412 12
23
Ax xx
bhx
++−
== +
2
32
234
234 12 12
xx
xxxx
+−
+++−
73. a. Substitute
3n=
into the formula:
3
30,000 30,000
1
x
x
−
−
b. Factor out 30,000 from the numerator:
33
30,000 30,000 1
30,000
11
xx
xx
−−
=
−−
c. Substitute in
1.05x=
into your formulas from
parts (a) and (b) above:
3
3
30,000 30,000
1
x
x
−
−
Total salary over three years is $94,575.
Chapter 5 Exponents and Polynomials
74. a.
4
30,000 30,000
1
x
x
−
−
b.
32
432
30,000 30, 000 30,000 30, 000
1 30, 000 0 0 0 30, 000
xxx
xxxxx
+++
−+++−
c. Let
1.08x=
.
()
4
4
30,000 1.08 30,000
30,000 30,000
11.081
$135,183.36
x
x
−
−=
−−
=
() () ()
32
32
30,000 30, 000 30,000 30, 000
30,000 1.08 30,000 1.08 30,000 1.08 30,000
37, 791.36 34,992 32, 400 30, 000 $135,183.36
xxx+++
=+++
=+++=
82. does not make sense; Explanations will vary. Sample explanation: The correct answer is
2
1.xx−+
83. makes sense
84. false; Changes to make the statement true will vary. A sample change is: The remainder is –9.
Section 5.6 Long Division of Polynomials; Synthetic Division
225
xx
=−+
89. Since the remainder is zero, we have
2
dividend
quotient divisor
16 2
quotient 21
xxk
x
=
−+
=−
90. Answers will vary. The quotient starts with x to a
power that is one less than the power in the
dividend. It is made up of terms that are all powers
of x down to 1, but with alternating signs. The
remainder is always
2−
. Following this pattern,
765432
12
1
11
xxxxxxx
xx
−=−+−+−+−
++
65432
765432
76
1
10000001
xxxxxx
xx x x x x x x
xx
−+−+−+
+++++++−
+
92. Let
2
1
25
5
x
yx
−
=− and
2
5yx=−.
The graphs do not coincide so the division is
incorrect.
2
5
5025
x
xx x
+
−+−
5
same set of axes.
The graphs do not coincide so the division is
incorrect.
The division leaves a remainder of 130.
The right side should be 130
223 .
5
xx
++
−
94. Let
2
1
6168
32
xx
yx
++
=+ and
2
24yx=−.
The graphs do not coincide so the division is
incorrect.
Chapter 5 Exponents and Polynomials
96. 7617
3 18
xy
xy
−=
+=
The addition method is a good choice because both
Back-substitute 5 for x in either equation of the
original system. We choose the original second
equation:
()
318
35 18
15 18
3
xy
y
y
y
+=
+=
+=
=
Solution: {(5,3)}
97. 6% 0.06, 20PB== =
98. 22
35 55
xx
+=−
To clear fractions, multiply by the LCD, 15.
The solution set is {−6}.
99. a.
3
5
77
7=7⋅7⋅
7 7⋅7⋅
2
1
77 7
=
⋅⋅
100.
34 4 34 12 12 10 2
10 10 10
(2 ) 2 ( ) 16 16 16
xxx
xx
xxx
−
====
1. a.
2
2
11
636
6
−
==
b.
3
3
11
5125
5
−
==
c.
4
4
11
(3) 81
(3)
−
−= =
−
d.
4
11
381
−
−=−=−
2. a.
23
2749
8
72
−
==
22
−
3.
12 2 12 2 10
10
1
xxx x x
−−+−
⋅= = =
4. a.
2210 8
10 8
1xxx
xx
−−
===
25
yy y
−
Section 5.7 Negative Exponents and Scientific Notation
5.
42 2 42 42 8
11 11 11 11
(6 ) 6 ( ) 36 36
xxxx
xxxx
⋅
===
7. a. The exponent is positive so we move the
decimal point eight places to the right.
9
7.4 10 7, 400, 000, 000×=
9. a.
82 82
82
10
(3 10 )(2 10 ) (3 2) (10 10 )
610
610
+
××=×××
=×
=×
10.
12 12 4
88
2.6 10 2.6 10 0.83 10 8300
3.12
3.12 10 10
×=×≈×=
×
Each citizen would have to pay about $8300.
5.7 Concept and Vocabulary Check
1.
1
n
b
5. true
6. false
9. false
5.7 Exercise Set
3
5
4.
3
3
11
464
4
−
==
8.
2
2
749
7
−
−=−=−
9.
1
1
11
44
4
−
==
10.
1
1
11
66
6
−
==
11.
11
11
1111
23 23
23
−−
+=+=+
Chapter 5 Exponents and Polynomials
13.
2
2
139
3
−
==
14.
3
3
1464
4
−
==
18.
32
23
4241
64 16
24
−
−
===
19.
222
22
11416
16
41
41
−−
−
====
23.
55
5
11
66
6
xx
x
−
⋅
==
24.
6
6
1
8
8
x
x
−
=
28.
() () ( )
3
3
443 427 108
3
−
=⋅− =⋅− =−
−
29.
83 83 5
5
1
xx x x x
−−+−
⋅= = =
33.
339 6
96
1xxx
xx
−−
===
34.
5512 7
12 7
1xxx
xx
−−
===
5
38.
4412 8
12 8
45 45 3
3
15
15
zzz
zz
−−
===
39.
33
4
77 4
88 4
4
xx
x
−
−−
= ⋅ =− =−
Section 5.7 Negative Exponents and Scientific Notation
43.
55
8
13 13 8
77 7 7
55
55
ww
w
ww w
−
=⋅ = =
47.
()
33
38 11
28 11
4
1yy
yy
yy
y
−−
−− −
== = =
48.
()
55
56 11
26 11
3
1yy
yy
yy
y
−−
−− −
== = =
51.
()
()
3
4312 12 5 17
55
66216 216
yyyy
yy
−−
−−
== =
52.
() ()
()
33
535 15
444
15 4 19
44 64
64 64
yy
y
yyy
yy
−−−
−−
==
==
x
55.
()
4
54
3412
2412
41
22
22
xxx
xx
−
−−−
===
57.
() ()
22
12122
22
2
33 3
9
3
xxx
xx
−−
−−−−
==
==
58.
()
()
()
2
22
2
112
2
1
44 16
4
x
xxx
−−
−
−−
==⋅=
3
33
y
−−
27 27
−
61.
57 12 12
666
6
6
23 6 6
15
15 15
22
55
xx x x
xxx
x
x
⋅==⋅
=⋅ =
62.
314 31414 3
14
35 35 3
20 4
20
xx xx
x
+−
⋅⋅
=⋅ =
Chapter 5 Exponents and Polynomials
66.
() ()
()
33
47347 127
12 7 5
33 27
27 27
yy yy yy
yy
−−−
+−
==⋅
==
70.
()
5
20 100
100
1
yy
y
−−
==
73.
()
8
4
26 8 24
24
a
ab ab b
−
−−
==
74.
()()()
555
72 7 2
35
35 10
10
ab a b
a
ab b
−−−
−−
−
=
==
78.
()
()
4
43
3128
24812
2
x
xxy
yyx
y
−
−−
−−
===
82. The exponent is positive so we move the decimal
point four places to the right.
4
7.24 10 72, 400×=
point one place to the left.
1
8.6 10 0.86
−
×=
87. The exponent is negative so we move the decimal
point two places to the left.
2
2.15 10 0.0215
−
×=
88. The exponent is negative so we move the decimal
point two places to the left.
Section 5.7 Negative Exponents and Scientific Notation
93.
8
220, 000, 000 2.2 10=×
94.
11
370,000, 000,000 3.7 10=×
95.
2
713 7.13 10=×
102.
6
0.00000103 1.03 10
−
=×
103.
3
0.005 5 10
−
=×
104.
3
0.006 6.0 10
−
=×
105.
0
3.14159 3.14159 10=×
109.
()()
53 53 8
9
2 10 8 10 16 10 16 10
1.6 10
+
× × =× =×
=×
112.
20 20
10 10
20 10
10
20 10 20 10
10
10 10 10
210
210
−
×=×
×
=×
=×
116.
22
33
23
5
18 10 18 10
9
910 10
210
210
−−
−−
−
×=×
×
=×
=×
663 3
180 10 90 10 90 10
−
×=× =×
5
910
=×
119.
443 7
3
310 0.25 10 0.25 10
12 10
+
−
×=× =×
×
Chapter 5 Exponents and Polynomials
121.
()
()
323
23 6
8
510 5 10 12510
1.25 10
×=×=×
=×
123.
()
()
424
24 8
7
3 10 3 10 81 10
8.1 10
−
−−
−
×=×=×
=×
124.
()()
55
35 3
15
115
14
210 2 10
32 10
3.2 10 10
3.2 10
−−
−
−
−
×=×
=×
=××
=×
127.
()
()
()
3
263
363
21
33
66 0 0
1
xy xy
xy
xy
xy xy
−
−−
−
−
−−−
−
=
===
128.
()
()
2
224
363
2
xy xy
xy
xy
−
−−
−−
−
=
130.
()
()
3
47 4733
27 7
277 77
3333
339
xyz x xyz x
yy
xyz xz xz
−
−− −−−−
−− −
=⋅⋅
===
132.
()
4
456 4
81012
456
32 40 48
32 40 48
1
xyz xy z
xyz
xyz xyz
−
−
−−−
−−−
=
==
133.
()()()
()
220
121 43 33
2
35
2216
2
xy xy xy
xy
−−
−−− − −
−−
()
26 6
4
xy
=
135.
()( )
()
34
3
2
510 1.210
2.5 10
2.4 10
−
−
××
=×
×
136.
()( )
23
4
3
210 2.610
1.3 10
−
−
××
=×
Section 5.7 Negative Exponents and Scientific Notation
138.
()( )
()()
62
1
63
1.2 10 8.7 10
1.2 10
2.9 10 3 10
−
−
××
=×
××
140. a.
12
1.09 10×
b.
4
2.5 10×
c.
12 12
44
8
7
1.09 10 1.09 10
2.5
2.5 10 10
0.44 10
4.4 10
44,000, 000 times
×=×
×
≈×
=×
=
143.
5
5
2.325 10 1.25
1.86 10
×=
×
seconds
144.
drt=⋅
so
d
tr
=
145. – 151. Answers will vary.
152. does not make sense; Explanations will vary.
Sample explanation:
3 9 27 12
36( ) 36 36xxx=≠
157. true
158. false; Changes to make the statement true will vary.
A sample change is:
4
(2) 16−=
and
4
4
11
2.
16
2
−
==
159. false; Changes to make the statement true will vary.
A sample change is:
22 22 0
55 5 5 1
−−
⋅= ==
and
55 55 0
22 2 2 1
−−
⋅= ==
()( )
()( )
()
53
55
5
5
710 210
7 10 0.00000002 10
7 0.00000002 10
7.00000002 10
−
×+×
=× + ×
=+ ×
=×
Chapter 5 Exponents and Polynomials
164.
12
12
11
22 22
11
24
−−
+=+
=+
165. The calculator verifies your results.
166. The calculator verifies your results.
167. The calculator verifies your results.
168. The calculator verifies your results.
169.
86 4 12
886 4 128
xx
xx
−>−
−− > − −
170.
() ()
()
24 8 3 28 7 3 3 28 7
945
÷⋅+ ÷− =⋅+ ÷−
=+− =
171. The whole numbers in the given set are 0 and
()
16 4=
.
Chapter 5 Review Exercises
1.
4
79xx+
is a binomial of degree 4.
2.
2
35 2xx+−
is a trinomial of degree 2.
5.
()( )
32 32
9754 710yy yyy−++ −+−
()( )
()
33 22
32
94 7 7510
13 8 7 5
yy yy y
yy y
=++−−++−
=−+−
6.
()( )
22
58634yy y y−− −− + −
()( )
22
58634
yy y y
=−−+−+
()
()()
()
432
44 33
22
43
5 3 2 6
13 5 8 3
2 2 6
856
xxx
xx xx
xx
xx
+− + − +
=−+−+
+−+
=−+