Chapter 5
Exponents and Polynomials
5.1 Check Points
1.
32 32
32 32
( 11 7 11 5) (16 3 3 15)
11 7 11 5 16 3 3 15
xx x xxx
xx x xxx
−+−−+ −+−
=− + − − + − + −
3.
22
22
22
2
(9 7 2) (2 4 6)
972246
927426
7114
xx xx
xx xx
xxxx
xx
+−− −−
=+−−++
= − ++−+
=++
6. Make a table of values.
()
()
2
2
1(,)
3(3)183,8
3(3)183,8
xyx xy
y
y
=−
−=−−=−
=−=
7. greatest
8. like
9. opposite
2.
52x− is a binomial of degree 1.
Chapter 5 Exponents and Polynomials
3.
2
2xx− is a binomial of degree 3.
4.
5
7xx− is a binomial of degree 5.
9.
2
34xx−+ is a trinomial of degree 2.
10.
2
92xx−+ is a trinomial of degree 2.
11.
24
795yy−+ is a trinomial of degree 4.
17.
()( )
()()
98 175
98(17)5
98175
917 85
813
xx
xx
xx
xx
x
++− +
=++− +
=+− +
=− ++
=− +
20.
()( )
22
22
22
11 7 4 27 10 20
11 7 4 27 10 20
xx x x
xx x x
+−+ + −
=+−+ +−
22.
()()
()
()
22
22
22
2
348
348
34 8
9
xx x x
xxx x
xx xx
xx
−++ +
=− + + +
=− + + +
=+
()
()()
22
22
2
7832 8
72 8 38
5911
xx xx
xx xx
xx
=− + + + + +
=− + + + + +
=− + +
25.
()( )
32
4751063yy yy+−+ −+
32
Section 5.1 Adding and Subtracting Polynomials
27.
()()
()
23
23
32
32
26733
26733
32 637
3297
xx xx
xx xx
xx xx
xxx
=−++−
=−++−
=++−−+
=+−+
29.
()( )
()
()()
23
23
32
32
4811 252
48112 52
24 85 112
241313
yy yy
yy yy
yy yy
yy y
++ +− ++
=+++− ++
=− + + + + +
=− + + +
33.
()( )
()
32 32 33 22
3
3
14 14
99
33 33
3
10 3
10 1
xxx xxx xx xx xx
x
x
−−− + +++ = + +−+ +−++−+
=+
=+
Chapter 5 Exponents and Polynomials
35.
()
432 432
44 2322
113 321
66
538 532
13 1231 66
55 3382
xxx xxx
xx xxxx
++++−+−−
=+− ++++− ++−
43 2
21
58
xx x=− + −
37.
()( )()
()( )
53 54 5 543
54 3
0.03 0.1 0.03 0.02 0.7 0.3 0.03 0.02 0.1 0.07 0.03 0.3
0.01 0.1 0.3 0.33
xxx xx x x xxxx x
xx x x
−+++− +−+= − +−+− ++
=+−++
40.
42
42
42
13
72
20
xx
xx
xx
−
+
+
Section 5.1 Adding and Subtracting Polynomials
43.
43
12
5
43
xx−−
43
11
4.7
25
xx−++
44.
95
95
11
2.7
35
32
1
43
xx
xx
−−
−++
45.
32
573yyy+−−
32
23411yy y−+ +−
32
8314yyy−+ − −
49.
432
32
43
73
4 2
7 2 4 2
xxx
xx x
xx x
−+
−+−
−+−
875
xx
+−
52.
2
2
2
7116
8 34
9 52
yy
yy
yy
−−
++
−−+
32
2
32
6.1 2.2 7
4.3 5
14 4.7 5.3
xx
x
xx
−+
−
−+
Chapter 5 Exponents and Polynomials
58.
()( )
22
3821154
8136
xx xx
xx
−−− ++
=− − −
59.
()( )
22
564xx xx−− −
()( )
()
()
22
22
2
564
654
5
xx xx
xx xx
xx
=−+−+
=− +−+
=− −
61.
()( )
22
89 5 43xx xx−−− −−
()( )
22
2
89 5 43
446
xx xx
xx
=−−+−++
=− − −
63.
()( )()( )
832 8 32
26
yyy y
y
−− − = −+−+
=− −
66.
()()
32 2
538811
5519
yy y y y
yy
+−−− −+
=+−
67.
()( )
37 37
78610nn nn−−− −−
()( )
()()
()
37 37
33 77
3
78610
76 810
2
nn nn
nn nn
n
=−−+−++
=−+−++−+
=+
69.
()()
63 2
yy yy−−−
()( )
63 2
632
yy yy
yyyy
=−+−+
=−−+
71.
()( )
745 195xxx xxx++−− −−
()()
42 42
42
745 195
xxx xxx
=+++ ++
Section 5.1 Adding and Subtracting Polynomials
73.
33
311 211
753 743
xx xx
−−−− +−
74. 22
22
22
2
2
311 111
834 824
311111
834824
31 11 11
88 32 44
423
866
15
26
xx xx
xx xx
xx xx
xxx
xx
−−−− +−
= −−+ −+
=++−−+−+
=+−−
=−
76.
()
42
35
x
x
+
−−
To subtract, add the opposite of the polynomial
being subtracted.
42
35
7
x
x
x
+
−+
+
78.
2
2
9 6
52
y
y
−
−− +
()
2
11 2 3
yy
−+−
To subtract, add the opposite of the polynomial
being subtracted.
2
2
2
7 5 2
11 2 3
475
yy
yy
yy
−+
+− − +
−−+
80.
()
53
53
3 5 6
742
xx
xx
−+
−+−
81.
()
32
265
xx
−− − +
To subtract, add the opposite of the polynomial
being subtracted.
32
32
32
753
265
9118
xx
xx
xx
+−
+−
+−
42
Chapter 5 Exponents and Polynomials
84.
()
32
32
4 5 7 11
5693
yyy
yyy
+++
−− + − −
To subtract, add the opposite of the polynomial
being subtracted.
()
32
32
32
45711
5693
9 16 14
yyy
yyy
yy y
+++
+−++
−+ +
86.
()
632
32
5 3 2
1
yyy
yyy
−+
−−−−−
To subtract, add the opposite of the polynomial
being subtracted.
()
63 2
32
632
5 3 2
+ + 1
5 2 3 1
yyy
yyy
yyyy
−+
+++
−+++
89. Table of values:
()
()
()
()
()
()
2
2
2
2
2
2
2
(, )
3(3)93,9
2(2)42,4
1(1)11,1
0(0)00,0
1(1)11,1
2(2)42,4
xyx xy
y
y
y
y
y
y
=
−=−=−
−=−=−
−=−=−
==
==
==
90. Table of values:
()
()
()
()
()
2
2
2
2
2
2
2
2(,)
3(3)273,7
2(2)222,2
1(1)211,1
0(0)220,2
1(1)211,1
xyx xy
y
y
y
y
y
=−
−=−−=−
−=−−=−
−=−−=−−−
=−=− −
=−=− −
Section 5.1 Adding and Subtracting Polynomials
91. Table of values:
()
()
2
2
2
1(,)
3(3)1103,10
2(2)152,5
xyx xy
y
y
=+
−=−+= −
−=−+=−
92. Table of values:
()
()
2
2
2
2(,)
3(3)2113,11
2(2)262,6
xyx xy
y
y
=+
−=−+= −
−=−+=−
93. Table of values:
()
()
2
2
2
4(,)
34(3)53,5
24(2)02,0
xyx xy
y
y
=−
−=−−=−−−
−=−−=−
94. Table of values:
()
()
2
2
2
2
9(,)
39(3)03,0
29(2)52,5
xyx xy
y
y
=−
−=−−=−
−=−−=−
Chapter 5 Exponents and Polynomials
96.
()()()
32 32 3
32 3
32
10 5 4 3 3 4 7 5 4
13 3 3 7 5 4
681
xx x xxx xx
xx x x x
xx x
−++−−−+− −+
=−++−+−
=−+−
99.
()( )( )
()
32 3 3 2
32 3 2
32
47322
373 22
2375
xx x x x x
xx x x x
xxx
++−+−−− +
=++−+−+−
=++−
102.
()( )()
()
23 3 23
32 2 3
32
65 1213 28
8628
16 3 6
yy yy yy
yyy y y
yyy
−+ + −− − + −− +
=− + + + + −
=− + + +
103. a.
()
32 32
32 32
0.2 1.5 3.4 25 0.1 1.3 3.3 5
0.2 1.5 3.4 25 0.1 1.3 3.3 5
Sx x x x x x
Sx x x x x x
=−+++ −++
=−+++−++
Section 5.1 Adding and Subtracting Polynomials
104. a.
()
32 32
32 32
0.02 0.4 1.2 22 0.01 0.2 1.1 2
0.02 0.4 1.2 22 0.01 0.2 1.1 2
Sxxx xxx
Sxxx xxx
=− + + + + − − + +
=− + + + − − + +
c. (5,36.75)
105. – 111. Answers will vary.
112. does not make sense; Explanations will vary. Sample explanation: When adding like terms, the exponents do not
change.
116. true
117. false; Changes to make the statement true will vary. A sample change is: The expression
2
11
3
5x
x
+ is not a polynomial
expression (variables must not be in the denominator).
118. true
xx
119. false; Changes to make the statement true will vary. A sample change is: The coefficient of x is –5.
120.
()
22
5213 2xx xx−+−− −−
Chapter 5 Exponents and Polynomials
122. Explanations will vary. A sample explanation is: In
a polynomial of degree 3, the highest degree term
has an exponent of 3. The highest degree term of the
sum will be the sum of two terms of degree 3,
which will be a term of degree 3 or could be 0. It is
impossible to get a term degree of 4, so it is
impossible to get a polynomial of degree 4.
125.
()()
3292
36918
369 9189
6618
666186
xx
xx
xxx x
x
x
−= +
−= +
−− = + −
−−=
−−+=+
b.
()
10
9 9 10 90
xxx
⋅
==
c.
3
77321
(5) (5) (5)
⋅
−=−=−
444 4
b.
45 45 9
(5 )(4 ) (54)( ) 20xx xx x−=−⋅⋅=−
5. a.
2
3( 5) 3 3 5 3 15xx x x x x x+= ⋅+⋅= +
b.
23
6(5 2 3)
xx x
−+
Section 5.2 Mullitplying Polynomials
8.
32
254
xx x
−+
5.2 Concept and Vocabulary Check
1.
;
mn
b
+
add
2.
;
mn
b
multiply
5.2 Exercise Set
1.
15 3 15 3 18
xx x x
+
⋅= =
2.
12 4 12 4 16
xx x x
+
⋅= =
7.
9 10 9 10 19
77 7 7
+
⋅= =
8.
710 710 17
88 8 8
+
⋅= =
3
3339
⋅
16.
()
333 3
44 64xxx=⋅=
17.
()()
22
22
5525xxx−=− =
18.
()()
22
22
6636xxx−=−⋅=
21.
()
()
()
44
4
6624
22 16yyy−=− =
22.
()
()
()
44
4
555420
2 2 16 16yyyy
⋅
−=−⋅= =
23.
()
()
()
55
5
7735
22 32xxx−=− =−
11 2
24 24
xx
+
==
27.
()
()
()
()
223
64 64 24xx xx x=⋅ ⋅ =
Chapter 5 Exponents and Polynomials
31.
()
32 32
5
11 11
24 24
1
8
aa aa
a
−−=−⋅−⋅
=
34.
()
()
()
()
()
36
36
316 10
325
325
30 30
xxx
xxx
xx
++
−
=⋅−⋅ ⋅⋅
=− =−
37.
()
2
33
3
xx xx x
xx
−=⋅−⋅
=−
40.
()
11
2
35335
335
315
xx xx x
xx
xx
+
−=⋅−⋅
=−⋅
=−
42.
()
11
2
56 7 56 57
56 57
30 35
yy yy y
yy
yy
+
−+=−⋅−⋅
=− ⋅ − ⋅
=− −
43
26
yy
=+
46.
()
22 22 2
22 2
42442
442
yy y yy y y
yyy
+
+=⋅+⋅
=+⋅⋅
48.
()
22
45 63
yy y
−+
43
68
xx
=+
50.
()
()
32 3 2
32
4524252
42 52
xxx xxxx
xx xx
+=⋅+⋅
=⋅ ⋅+⋅ ⋅
Section 5.2 Mullitplying Polynomials
53.
()
22
34 5xxx−−+−
()
() ( )
22 2 2
43 2
34 3 35
12 3 15
xx xxx
xx x
=− − − − −
=−+
55.
()()
35xx++
()()
2
2
53 5
53 35
5315
815
xx x
xx x x
xxx
xx
=+++
= ⋅+⋅+⋅+⋅
=+++
=++
57.
()()
21 4xx++
()()
2
2
2414
28 4
294
xx x
xxx
xx
=+++
=+++
=++
2
91199
299
xxx
xx
=+− −
=−−
62.
()()
()()
12 8
812 8
xx
xx x
−+
=+− +
2
2
28520
2320
xxx
xx
= +−−
=+−
64.
()()
()()
2
34 5
3545
335445
xx
xx x
xx x x
−+
=+−+
= ⋅+ ⋅−⋅−⋅
()
()
14 1
44 4
13 1 1
44 4
3
+4 4 1
4
xx x
xx x
x
=−+−
=⋅+ −
+−
Chapter 5 Exponents and Polynomials
66.
13
51
55
25 5
xx
+−
67.
()
()
2
123xxx+++
68.
()
()
()()
22
22
32 2
32
25
52 5
52 2 25
52 210
3710
xxx
xx x x x
xx xx x x x
xx xx x
xxx
+++
=+++++
=⋅ +⋅+⋅+⋅ +⋅+⋅
=+++ ++
=+ ++
70.
()
()
()()
2
22
22
32 2
32
243
432 43
4322423
43286
6116
yyy
yy y y y
yy y y y y y
yyyyy
yy y
−−+
=−+−−+
=⋅ −⋅ +⋅−⋅ +⋅ −⋅
=− +− +−
=− + −
72.
()
()
2
22
21 43
aaa
−−+
()()
()( )
()
432 32
433 22
234 234
232434
xxxxxxx
xxx xx xx
=+ + +++ ++
=+ ++ + +++
432 32
43 2
473 473
511103
xxxxxxx
xx x x
=+ + +++ ++
=+ + + +
75.
()
32
14256
2
xxxx
−−+−
1
76.
()
32
13659
3
xxxx
−−+−
()()
32 32
432 32
432
1
3659 3659
3
5
3659 2 3
3
32
377 3
3
xx x x x x x
xxxxxx x
xxx x
= − +−− − +−
=−+−−+−+
=−+− +
Section 5.2 Mullitplying Polynomials
78.
()()
()()()
22
22 2 2
43232 2
43 2
31 21
213 211 21
236321
651
xx xx
xxx xxx xx
xxxxxxxx
xx x x
++ −−
=−−+−−+−−
=− −+ − −+−−
=+− −−
80.
2
2
32
32
79
4
42836
79
31936
xx
x
xx
xx x
xx x
−+
+
−+
−+
−−+
292727
xx x
−+−
82.
2
2
32
32
5 3
4 5
5 25 15
42012
4253715
yy
y
yy
yyy
yyy
−+
−
−+−
−+
−+−
85.
32
32
42 54
3 2
84108
zz z
z
zz z
−+−
−
−+ − +
432
432
12 5 15 12
12 14 19 22 8
zzzz
zzzz
−+−
−+−+
10 26 20 22 12
zzzz
−+−+
87.
32
2
432
7 5 6
3 4
28 20 24
xxx
xx
xxx
−+
−
−+ −
543
5432
21 15 18
xxx
−+
5432
27 24 20 25
yyyy
+−+
89.
532
532
6432
2 3 2 3
2 1
2 3 2 3
4 6y 2 4 6
yyyy
y
yyyy
yyyy
+
−+−+
−
−−+−
−+−+
Chapter 5 Exponents and Polynomials
91.
2
2
2
73
1
73
xx
xx
xx
+−
−−
−− +
32
432
43 2
7 3
7 3
61143
xxx
xxx
xx xx
−− +
+−
+− −+
93.
()()()()
()( )
()( )
22
22
45 36
20 3 18
20 3 18
22
xx xx
xx x x
xx x x
x
+−−+−
=−−−−−
=−−+−++
=−
15 42 8
xxx
=+−
96.
()()
()()
23 32
532 5 3
532
36 234 5
18 6 9 4 20
14 26 9
xx x xx
xxx x x
xxx
+−− −
=+−+−+
=+−
98.
()
()
()
()
()()
()()
22
22
22
32 2
32 2
1111
11 1
1 1 1
1
1
2
yyy yyy
yy y y y
yy y y y
yyyyy
yyyyy
+−+−−++
=−++−+
−+++++
=−++−+
−−−+++
=
100.
()()
()()()()
()()
22
54
55 44
10 25 8 16
18 9
yy
yy yy
yy yy
y
+−−
=+ +−− −
=++−−+
=+
101. Use the formula for the area of a rectangle.
()( )
()( )
()
2
2
14310
2
23 10
23 210
A
xx
xx
xx x
=
=+
=+
=⋅+⋅
Section 5.2 Mullitplying Polynomials
104. a. The length of the larger rectangle is
23x+
and
the width is
2x+
. Therefore, the area of the
rectangle is given by
()()
23 2
Alw
xx
=⋅
=+ +
The area of the larger rectangle is
()()
23 2xx++
square feet.
105. When multiplying numbers with the same base,
keep the base and add the exponents.
Example:
35 35 8
22 2 2
+
⋅= =
106. – 112. Answers will vary.
113. makes sense
120. false; Changes to make the statement true will vary.
A sample change is: It is not necessary to use the
vertical format for any polynomial multiplication.
121. The area of the outer square is
()()()()
2
44 444
4416
xx xx x
xxx
++=+++
=+++
()()
22
32 2
3
11 1
1
1
xx x x x
xxxxx
x
=++−++
=++−−−
=−
c.
()
()
()()
32
32 32
11
11 1
xxxx
xx x x x x x
−+++
= + ++ − + ++
8.x−
Chapter 5 Exponents and Polynomials
125. 32 6xy−=
x-intercept:
32 6
32(0)6
36
2
xy
x
x
x
−=
−=
=
=
A checkpoint is (4,3).
126.
()
21
22
68 2
12 3
yy
mxx
−−
== =−
−−−
2
2
339
69
xxx
xx
=+++
=++
b.
2
2
2
(5) (5)(5)
559
10 25
xxx
xxx
xx
+=+ +
=+++
=+ +
2.
2
2
(7 5)(4 3) 7 4 7 ( 3) 5 4 5( 3)
28 21 20 15
28 15
xx xxx x
xxx
xx
+−=⋅+−+⋅+−
=−+−
=−−
2
2
20 22 6
62220
xx
xx
=− +
=−+