Mid-Chapter Check Point
3.
()( )
3547
xx
+−
5.
()
()
()()
2
22
32 2
32
25 31
231531
2625155
211175
xxx
xx x x x
xxxx x
xx x
−−+
=−+−−+
=−+−+−
=− +−
8.
()()
45 9
10 7 70xx x−−=
9.
()()
()
22
2
22
4
22
2
4
xx
x
x
+−
=−
=−
11.
()()
22
9102
18 9 20 10
abab
aabbab
−+
=+−−
13.
23 2 23 2
34235
ab ab b ab ab b
−+ −− − +
15.
()
()
()()
()()
32
32
33 22
9527
11 6 3 7
911 56
23 77
xxx
xxx
xx xx
xx
−+ −+
+−+−
=− + + −
+− + + −
()
()
()
54
54
54
54
26713
2310
26713
471023
xxx
xx x
xxx
xx x
−−+−
=− + − +
+− + − +
=− + − +
19.
()()
43
743
5421
10584
xx
xxx
+−
=−+−
Chapter 5 Exponents and Polynomials
22. Make a table of values:
()
()
()
2
2
2
2
1(,)
31(3)83,8
21(2)32,3
11(1)01,0
xyx xy
y
y
y
=−
−=−−=−−−
−=−−=−−−
−=−−= −
5.5 Check Points
1. a.
12 12 4 8
4
555
5
−
==
2. a.
0
14 1=
b.
0
(10) 1−=
c.
4
10 4 10 4 40
33412
22()16
()
aaa
bbb
==
4. a.
12 12 4 8
4
20 20 2
10
10
xxx
x
−
−−
==−
444 0
5
7.
76 23 2 76 23 2
2222
64
1866018 6 60
6666
310
xy xy xy xy xy xy
xy xy xy xy
xy xy
−+ =−+
=−+
3.
;
n
n
a
b
numerator; denominator
4. divide; subtract
Section 5.5 Dividing Polynomials
5.5 Exercise Set
1.
20 20 5 15
5
333
3
−
==
4.
884 4
4
xxx
x
−
==
5.
13 13 5 8
5
yyy
y
−
==
9.
100 50 100 25 50 10 75 40
25 10
xy xy xy
xy
−−
==
10.
200 40 200 25 40 10 175 30
25 10
xy xy xy
xy
−−
==
11.
0
21=
19.
()
0
100 1y=
20.
()
0
200 1y=
0
=
23.
()
0
011 2
ππ
−−− =−−=−
24.
()
0
0
3311112−−− =−−=−−=−
27.
33
464
44
===
28.
()
3
32
2236
3
32727
3
x
xxx
⋅
===
29.
()
2
223
36
2
2
24
525
5
x
xx
==
Chapter 5 Exponents and Polynomials
33.
()
()
()
()
()
55
5
577
7
455
44
35 35
20 20
22
2
32 32
aa
a
bbb
aa
bb
−−
−==
−
==−
35.
()()
44
423
23 812
44 4
2216
xy
xy xy
zzz
==
38.
12 12 4 8
4
45 45 3
15
15
xxx
x
−
==
39.
22 22 2 20
2
88 2
4
4
xxx
x
−
−−
==−
42.
13 13 9 4
9
15 15 1
45 3
45
yyy
y
−
−−
==−
45.
75 72 51 5 4
2
30 30 6
5
5
xy xy xy
xy
−−
==
46.
95 92 51 7 4
2
40 40 20
2
2
xy xy xy
xy
−−
==
2
48.
16 2 16 2 2 2
22
15 15
45
45
xy xy
xy
−−
−−
=
50.
30 30
00
15
15
77
15 15
xy
xy
=
==
51.
10 12 6 10 2 12 3 6 2
232
51
10
50
xyz xyz
xyz
−−−
−=−
10 7 2
1
5
xyz
=−
53.
43 4 3 43
10 2 10 2 5
xx x x xx
+=+=+
Section 5.5 Dividing Polynomials
55.
43 4 3
41 31 3 2
14 7 14 7
777
22
xx x x
xxx
xx xx
−−
−=−
=−=−
57.
72 7 2
71 21 11
610
6
99
9
9
91
yyyy yy
yyyy
yyy
yyy
yy
−−−
−+
=− +
=− +
=− +
=−+
59.
32 3 2
31 21 2
24 15 24 15
333
85 85
xx x x
xxx
xx xx
−−
−−
=+
−−−
=− + =− +
61.
543 5 4 3
2222
32
18 6 9 18 6 9
3333
623
xxx x x x
xxxx
xxx
++ =++
=++
63.
43 2 4 3 2
32
12 8 40 12 8 40
4444
3210
xx x x x x
xxxx
xx x
−+ =−+
=−+
32
777
7210
xxx
xx x
=+ +
−−−
=− + −
65.
()
22
2
464 6
46
46
xxx x
xxx xxx
x
−
−÷= = −
=−
16 8
y
=−
67.
32 3 2 2
30 10 30 10 62
555
zz z z zz
zzz
+=+=−−
−−−
69.
32 3 2
2
8628 6 2
2222
431
xxxx x x
xxxx
xx
+−
=+−
=+−
Chapter 5 Exponents and Polynomials
71.
754 7 54
3333
42
25 15 5 25 15 5
5555
53
xxx x xx
xxxx
xxx
−−
=−−
=−−
73.
76 5 4
4
18 9 20 10
2
xx x x
x
−+ −
−
76 5 4
4444
32
1892010
2222
9
9105
2
xx x x
xxxx
xxx
=−+−
−−−−
=− + − +
75.
22 2 2
12 6 15
3
xy xy xy
xy
+−
77.
74 32 2
2
20 15 10
5
xy xy xy
xy
−−
−
74 32 2
20 15 10
xy xy xy
−−
66 32 86 22 146 62
01 20 84
284
23
23
23
xy xy x y
xy xy xy
yxxy
−− −− −−
=− + +
=− + +
=− + +
79.
()()
32
2
433 2
242324
2
84612
xx xx
x
xxx x
+− −
+−+
2
432
2
3
18 24 15
x
xxx
+−
=
Section 5.5 Dividing Polynomials
81.
24 56 22
244
2
18 15 23
95
xy xy xy xy
xy x y
xy
−=−
=−
83.
()()()
()()
2
22
22
555
2
10 25 25
2
2102 10 5
222
yyy
y
yy y
y
yyy y
y
yyy
+++ −
+++−
=
+
==+=+
85.
15 12 3
3
15 12 3
333
12 9
12 24 8
4
12 24 8
444
362
nnn
n
nnn
nnn
nn
xxx
x
xxx
xxx
xx
−+
=−+
=−+
87. a.
Receipts 5022
Average 4.22
Admissions 1189
==≈
The average admission charge for a film in 1990 was about $4.22.
Chapter 5 Exponents and Polynomials
88. a.
Receipts 7468
Average 5.40
Admissions 1383
==≈
The average admission charge for a film in 2000 was about $5.40.
d. Polynomial division cannot be performed using the methods in this section because the divisor is not a monomial.
89. – 95. Answers will vary.
96. does not make sense; Explanations will vary. Sample explanation: Numbers to 0 powers equal 1.
100. false; Changes to make the statement true will vary. A sample change is:
10 2 10 2 8
xxx x
−
÷= =
101. false; Changes to make the statement true will vary. A sample change is:
33
2
12 6 12 6 63
222
xx x x
x
xxx
−=−=−
Section 5.5 Dividing Polynomials
104.
864
64 2
2
18 27 36 6912
3
xxx
xx x
x
−+ =−+
105.
86 53
?
?? 34
3
xx xx
x
−=−
To get 3 as the coefficient of the first term of the
quotient, the coefficient of the first term of the
106. To get 2 as the coefficient of the middle term of the
quotient, the coefficient in the divisor must be −3.
107.
20.3 20.3−=
108.
0.875
8 7.000
109.
12
3
yx=+
0
The quotient is 26 and the remainder is 0.
The quotient is 123 and the remainder is 6.
112.
257
98 25187
196
Chapter 5 Exponents and Polynomials
5.6 Check Points
1.
2
2
5
91445
9
x
xx x
xx
+
+++
+
2.
22
68 128 612
23 23
xx x x
xx
+− +−
=
++
2
43
238 612
x
xxx
−
++−
3. Rewrite
3
1x−
using coefficients of 0 on the
missing terms gives
32
001.xxx++−
2
32
32
1
1001
xx
xx x x
xx
++
−++−
−
2
2432
43
32
2714
22 3 0 7 10
24
7 0
xx
xxxxxx
xx
xx
++
−++−−
−
−
43 2
22
23710 2110
2714
22
xxx x
xx
xx xx
+−− −
=+++
−−
5.
()
()
32
76 2 23xx x xx−−÷+=−−
1.
32
10 4 0 9xxx+++
2.
2
8;x
2x; 4x; 10x
3. 5x; 3x 2;
2
15 10 ;xx−
2
15 22xx−
Section 5.6 Long Division of Polynomials; Synthetic Division
5.6 Exercise Set
1.
2
4
268
x
xxx
+
+++
2.
2
2
2
5710
5
2 10
2 10
0
x
xx x
xx
x
x
+
+++
+
+
+
2710 2
5
xx x
x
++=+
+
4.
2
2
23
5 2 13 15
2 10
x
xxx
xx
+
+++
+
5.
2
2
2
356
3
x
xxx
xx
−
−−+
−
6.
2
2
4224
4
6 24
6 24
0
xxx
xx
x
x
+−−
+
−−
−−
2224 6
4
xx x
x
−− =−
+
21
y
+
8.
2
2
27
3 2 13 21
2 6
y
yyy
yy
−
−−+
−
Chapter 5 Exponents and Polynomials
9.
2
5
234
x
xxx
−
+−+
10.
2
2
10
375
3
x
xx x
xx
−
+−+
+
11.
2
2
3
2510
2
3 10
3 6
4
y
yyy
yy
y
y
+
+++
+
+
+
22
510 510 4
3
22 2
yyyy y
yy y
++ + +
==++
++ +
13.
2
32
52
1672
xx
xx x x
−+
−−+−
32 2
672 52
1
xxx xx
x
−+−
=−+
−
2
23
3 3
3 3
0
xx
x
x
++
+
+
32 2
353 23
1
xxx xx
x
+++
=++
+
15.
2
2
61
2 3 12 20 3
12 18
2 3
y
yyy
yy
y
−
−−+
−
−+
Section 5.6 Long Division of Polynomials; Synthetic Division
17.
2
23
214 43
a
aaa
+
−+−
18.
2
2
5
212 95
2
10 5
10 5
b
bbb
bb
b
b
−
+−−
+
−−
−−
0
23
32 2
322
21
232 2
21
yy y
y
yy y yy
y
−+ +
+
−++
==−+
+
20.
2
32
356
4312 11 918
yy
yyyy
−+
+−++
23
2
91811 12 356
43
yyy y yy
y
+− + =−+
+
21.
2
2
35
256 530
x
xxx
+
−−−
8
2
483 8
25
21 21
yy y
yy
++
=++
−−
23.
2
32
28
2043
xx
xxx x
++
−++−
Chapter 5 Exponents and Polynomials
24.
2
32
32
48
2203
2
xx
xxx x
xx
++
−++−
−
25.
2
32
21
234 8 59
yy
yyyy
++
++++
32
4 6
yy
+
26.
2
32
32
2
212 32
2
yy
yyyy
yy
−+
+−++
+
27.
2
32
232
326 5 05
yy
yyyy
−+
+−++
32
6 4
32 3
yy
y
+
+2y+
28.
2
32
32
236
234 0 35
yy
yyyy
++
−+++
23 23
yy
−−
29.
2
32
931
3127 0 01
xx
xxxx
++
−++−
32
Section 5.6 Long Division of Polynomials; Synthetic Division
30.
2
32
469
238 0 027
xx
xxxx
−+
++++
31.
32
43 2
92727
3 12 54 108 81
yy y
yy y y y
−+−
−−+ −+
43
32
32
3
9 54
9 27
yy
yy
yy
−
−+
−+
32.
32
43 2
6128
2 8 24 32 16
yy y
yyy y y
+++
+++++
0
34 2
32
8163224
2
6128
yy y y
y
yy y
+++ +
+
=+ + +
33.
2
24
214 6 0
y
yyy
+
−++
Chapter 5 Exponents and Polynomials
35.
32
432
1
10205
yyy
yy y y y
+−−
−+−++
43
yy
−
42 32
25 4
1
11
yy yyy
yy
−+
=+−−+
−−
36.
32
432
55
10603
yy y
yy y y y
+−−
−+−++
32
432
55
10603
yy y
yy y y y
+−−
−+−++
2
−
42 32
63 2
55
11
yy yy y
yy
−+
=+−−−
−−
37.
()()
432 2
43496 3xxxx x+++−÷+
Rewrite the divisor with the missing power of x and
divide.
2
2432
438
0343 496
xx
xx xx xx
+−
++ + + +−
432
2
2
2
43496
3
18
438 3
xxxx
x
xx x
++ +−
+
=+−+
+
4 0 8
xx
+−
2
x
−
53 2 3
22
34128 2
354
22
xx x x x
xx
xx
−+ − −
=++−
−−
39.
()()
432 2
15 3 4 4 3 1xxx x+++÷ −
7
x
+
432 2
22
2
2
15 3 4 4 7
53
31 31
7
or 5 3 31
xxx x
xx
xx
x
xx x
+++ +
=+++
−−
+
+++
−
Section 5.6 Long Division of Polynomials; Synthetic Division
40.
2
2432
432
631
301189300
18 0 6
xx
xx xxxx
xxx
+−
++ + + ++
++
41.
()
()
2
2102xx x+− ÷ −
2 2 1 10
410
25 0
−
()
()
2
210225xx x x+− ÷ − = +
42.
1 1 1 2
−
()
()
2
20
3720 538 5
xx x x x
+− ÷+=−+
+
44.
3 5128
15 81
52773
−−−
−
−
45.
()
()
32
4331 1xxx x−+−÷−
1 4 3 3 1
−−
()
()
32
2
56311 2
33
5411 2
xxx x
xx x
−++÷−
=+++
−
47.
()
()
532
62431 2xxxx x−+−+÷−
2 602431
12 24 44 96 186
612224893187
−−
43 2
549
72160182 3
xx x x x
=+ + + + +
−
Chapter 5 Exponents and Polynomials
49.
()
()
234
55 5x xxx x−− + ÷+
Rewrite the polynomials in descending order.
32
1300
10 51 260 5
xxx x
=− + − +
+
50.
6 1 6 1 6 0
6 72 438 2664
1 12 73 444 2664
−− −
−−
−−
51.
()
32
1
3241 3
xxx x
+−+÷−
1 3 2 4 1
3111
33 3 0
−
−
−
53.
53
2
1
xx
x
+−
−
75 3
654 3 2
10 12
2
68
2 5 10 10 20 40 2
xx x
x
xxx x x x x
+− +
+
=− + − + − +−
+
432
256 41664
4
xxx x
x
−=+ + +
−
56.
21 0 0 0 0 0 0 128
248163264 128
1 2 4 8 16 32 64 0
−
Section 5.6 Long Division of Polynomials; Synthetic Division
58.
2 1 2 1 3 1 1
20222
101113
−− −
−
−
59.
32 23
432 4
43
32
000
0
xxyxy y
xyxxxxy
xxy
xy x
−+−
+++++
+
−+
60.
43 22 3 4
5432 4
54
43
432
0000
0
xxyxyxyy
xyx x x x xy
xxy
xy x
xy xy
−+ −+
++++++
+
−+
−−
61.
2
2432
432
321
23 5 7 3 2
3 3 6
xx
xx xxxx
xxx
+−
++ + + + −
++
2
0
xx−−−
432 2
2
35732
321
2
xxxx xx
xx
+++−
=+−
++
62.
2
2432
21
32 7 72
xx
xxxxxx
++
−− −− −−
63.
232
32
2
2
47
14 3 1
4 4 4
7 3 1
7 7 7
4 8
x
xx x xx
xxx
xx
xx
x
−
++ − ++
++
−−+
−−−
+
Chapter 5 Exponents and Polynomials
65.
32
2 5432
543
432
3
200001
2
2 0
xxx
xxxxxxx
xxx
xxx
+−−
−+ ++++−
−+
−+
66.
2
35432
52
43
573
4 5 7 3 20 28 12
5 20
7 3 28
xx
x xxx x x
xx
xx x
−+
−−+−+−
−
−+ +
67.
22
32 23
32
45
34 7 16 3
4 12
xxyy
xyx xy xy y
xxy
+−
−−−+
−
68.
22
32 23
32
22
32
4 5 12 19 13 10
12 15
xxyy
xyx xy xy y
xxy
−+
−−+−
−
()( )
32 3 2
32
4273274
353
xx x x x x
xxx
+−+− − −+
=+ ++
Now, complete the division:
32
353
1
xxx
x
+++
+
32 2
3 3
3 3
0
353 23
x
x
xxx xx
+
+
+++
=++