Section 5.3 Special Products
b. Since this product is of the form
()()
A
BAB+−
,
use the special-product formula
A
A
c. Since this product is of the form
()()
A
BAB+−
,
use the special-product formula
5. a. Use the special-product formula
22 2
() 2 .ABAABB+=+ +
6. a. Use the special-product formula
22 2
() 2 .AB A ABB−=− +
first term last term
2 product
squared squared
of the terms
22 2
2
(9) 29 9
18 81
xx x
xx
⋅
−= −⋅ +
=− +
5.3 Concept and Vocabulary Check
2
4.
22
2;
A
AB B−+
minus; product of the terms;
5.3 Exercise Set
2
12
yy
=+−
5.
()()
2
2
23 52 10315
2715
xx xxx
xx
−+=+−−
=+−
6.
()()
2
2
35 73 21535
xx x xx
−+=+−−
Chapter 5 Exponents and Polynomials
10.
()()
2
2
257214 43510
14 31 10
xx xxx
xx
−+=+−−
=−−
13.
()()
2
2
73 15 735 3 15
15 32 7
xx xxx
xx
+−=−+−
=− − +
16.
()( )
2
2
2
72 103
70 21 20 6
70 41 6
64170
yy
yyy
yy
yy
−−
=− − +
=− +
=−+
18.
()()
22
422
42
7235
21 35 6 10
21 41 10
xx
xxx
xx
−−
=−−+
=−+
21.
()
()
23 2
32
53 3515
5315
xx xxx
xxx
++=+++
=+ ++
25.
()()
22 2
33 3 9xx x x+−=−=−
29.
()()()
22
2
3434 3 4
916
rr r
r
−+=−
=−
30.
()()()
22
2
5252 5 2
zz z
−+=−
33.
()()
()
22 2
5 7 5 7 5 7 25 49xx x x−+=−=−
34.
()() ()
2
2
2
43 43 4 3
16 9
yy y
y
−+=−
=−
Section 5.3 Special Products
36.
()
2
2
2
11 1
33 3
33 3
1
99
yy y
y
+−=−
=−
40.
()()()
2
33 32
6
44 4
16
mm m
m
+−=−
=−
41.
()() ()
2
4424 8
11 1 1yy y y−+=− =−
44.
()()()
2
12 12 12 2
24
33 3
9
xx x
x
+−=−
=−
45.
() ()
222
2
2222
44
xxx
xx
+=+ +
=++
49.
() ()
222
2
3233
69
xxx
xx
−=− +
=−+
50.
() ()()
222
2
6266
12 36
xxx
xx
−=− +
=− +
51.
()()()()
22 2
34 3 2344
yyy
−= − +
54.
()()()
()
22
2222
42
53 5 2533
25 30 9
xxx
xx
−= − +
=−+
55.
() ()()()
22
2
2
72 7 272 2
49 28 4
xxx
xx
−=− +
=− +
2
1
42
4
xx
=++
58.
() ()
22
2
2
111
3323
333
1
92
9
xxx
xx
+= + +
=++
Chapter 5 Exponents and Polynomials
63.
()
()
2
11xxx−++
()()
22
32 2
3
11 1
1
1
xx x x x
xxxxx
x
=++−++
=++−−−
=−
65.
() ()()
222
2
1211
21
xxx
xx
−=− +
=−+
69.
()
22
34 9xxx++
()
() ()
22 2 2
43 2
34 3 39
12 3 27
xx xx x
xx x
=++
=++
21
xx
=+ +
74.
()()()
()
22
2222
42
2222
44
xxx
xx
+= + +
=+ +
75.
()( )
22
12xx++
77.
()()()
2
22 22
4
44 4
16
xx x
x
+−=−
=−
612
412 9
xx
=− +
81.
22
13
12 8
44
xx
+−
13 1 3
Section 5.3 Special Products
16
83.
()
22
121Ax x x=+ =++
84.
()()()
()()
2
22
2
33 3
233
69
xx x
xx
xx
++=+
=+ +
=++
87. Area of outer rectangle:
()()
2
93 1227xx xx++=++
Area of inner rectangle:
()()
2
51 65xx xx++=++
Area of shaded region:
()()
22
12 27 6 5 6 22xx xx x++−++=+
88. Area of outer rectangle:
()()
34
o
Ax x
=+ +
42
94
81 72 16
x
xx
=−
=−+
91.
()
()()
()
2
22
4
412121
4141
16 1
xxx
xx
x
++−
=+ −
=−
()()
()
()
()()
2
22
32 2
32
22
244
442 44
44288
6128
xx
xxx
xx x x x
xxxxx
xx x
=+ +
=+ ++
=+++++
=+ ++ ++
=+ + +
94.
()()
3
2
(4)
44
x
xx
+
=+ +
Chapter 5 Exponents and Polynomials
97.
()()
21xx++
The area of the larger garden is given by
()()
21xx++
square yards.
100. If the original garden measures
8x=
yards on a
side, the area of the larger garden would be
() ()
2
838290++=
square yards.
This corresponds to the point
()
8, 90
on the graph.
102.
()
()
frame total painting
22
222
2
22 2
44
AAA
xx
xxx
x
=−
=+ −
=+⋅ +−
=+
The area of the frame is
(4 4)x+
square inches.
109. makes sense
110. makes sense, although answers may vary
111. makes sense, although answers may vary
114. false; Changes to make the statement true will vary.
A sample change is: Since
22 4
,xx x⋅=
22 4
(3 2)(3 2) 9 4.xx x+−=−
117.
()()()
()
()
2
2
10 2 8 2
80 20 16 4
Vlwh
xxx
xxxx
=⋅ ⋅
=− −
=−−+
The volume of the box is
32
(4 36 80 )xxx−+
cubic units.
118. Divide the figure into two rectangles by drawing a
vertical line.
2
3
2
xx x
xx
=+−
=+
119. Let
()
2
1
1yx=+
and
2
2
1yx=+
.
The graphs do not coincide so the multiplication is
Section 5.3 Special Products
121. Let
()()
1
11yx x=+ −
and
2
2
1yx=−
.
The graphs coincide so the multiplication is correct.
()()
22
11 1 1xx xxx x+ −=−+−=−
123.
231
37
xy
yx
+=
=−
The substitution method is a good choice because the second equation is already solved for y. Substitute
37x−
for y
into the first equation.
()
{
}
124.
34 7
27 9
xy
xy
+=
+=
The addition method is a good choice because both equations are written in the form .
A
xByC+=
To eliminate x,
multiply the first equation by 2 and the second equation by −3. Then add the results.
6 8 14
xy
+=
Chapter 5 Exponents and Polynomials
125.
1
3
yx≤
126.
32 3 2
2 5 2 ( 2) (3) 2( 2)(3) 5( 2) 2
( 8)(3) 2( 2)(9) 5( 2) 2
24 36 10 2
72
xy xy x++−=− +− +−−
=− + − + − −
=− − − −
=−
5.4 Check Points
1. Begin by substituting
1−
in for x and 5 in for y.
32 3 2
3 5 6 3( 1) (5) ( 1)(5) 5(5) 6
3( 1)(5) ( 1)(25) 5(5) 6
15 25 25 6
9
xy xy y+++=− +− + +
=− +− + +
=− − + +
=−
Section 5.4 Polynomials in Several Variables
3.
22
22
( 8 3 6) (10 5 10)
83610510
xy xy xy xy
xy xy xy xy
−−++ +−
=− − + + + −
5.
342 432
14 32
55
(6 )(10 ) (6 10)( )( )
60
60
xy x y x x y y
xy
xy
++
=⋅ ⋅ ⋅
=
=
7. a.
O
FIL
22
22
(7 6 )(3 )
(7 )(3 ) (7 )( ) ( 6 )(3 ) ( 6 )( )
21 7 18 6
21 25 6
xyxy
xx x y yx y y
xxyxyy
xxyy
−−
=+−+−+−−
=−−+
=−+
Chapter 5 Exponents and Polynomials
5.4 Concept and Vocabulary Check
1.
18−
2. 6
5.4 Exercise Set
1.
()( ) ( )
2
222
222233
4129 1
xxyy++=+ −+−
=− +=
4.
()( ) ()( )
()()
3
3
22 3 232
83 62
24 6 2
16
xy xy−+= −− −+
=⋅− −− +
=− + +
=−
Section 5.4 Polynomials in Several Variables
7.
32 27 2
563xy xy y−+−
32
Term Coefficient Degree
1325
xy
+=
8.
4372
12 5 4xy xy x−−+
4
Term Coefficient Degree
12 12 4 1 5
xy
+=
9.
()()
22
53 2xy xy xy xy−+ −
()
()
22
2
52 3
74
xy xy xy xy
xy xy
=++−−
=−
12.
()( )()
()()
2222
2
751336473 56 134
41117
xy xy xy xy xy xy xy xy
xy xy
+++− ++= − + + ++
=++
13.
()( )
42 22 42 22
753 186
xy xy xy xy xy xy
−++− −−
Chapter 5 Exponents and Polynomials
15.
()()
3232
75 6 4xxyy xxyy+− − −+
16.
()( )
4343
75 634
xxyy xxyy
−− − −+
17.
()
42 3
353xy xy y+−
()
42 3
2346xy xy y x−−−+
()( )
()()
()()
42 3 42 3
42 42 3 3
42 3
3532346
32 53 346
86
xy xy y xy xy y x
xy xy xy xy y y x
xy xy y x
=+−+−++−
=−+++−++−
=++−
19.
()( )
()( )
()( )
33 32 2 3
33 32 2 3
33 332 2
332 2
32 2 3
43
43
43
54
54
xy xxyxy y
xy xxyxy y
xx yyxyxy
xyxyxy
xxyxy y
−−−− ++
=−+ + −−
=+ +−− + −
=−+−
=+−−
Section 5.4 Polynomials in Several Variables
22. Add:
22 2 2
22 2 2
22 2 2
7 5 6
10 6 6
312
ab ab b
ab ab b
ab ab b
−+
−++
−++
24.
()
24 2
24 2
13 17
78
xy xy xy
x y xy xy
−+
−− − −
To subtract, add the opposite of the polynomial being subtracted.
25.
()( )()
713 2619 115 713 2619115
7261113195
30 37
xy xy xy xy xyxy
xxxyyy
xy
++−+ −−=+−+−+
=− − + + +
=− +
28.
()
()( )
()
()
22
21 11
32
10 5 10 5
50
50
xy xy x x yy
xy
xy
++
=⋅ ⋅ ⋅
=
=
Chapter 5 Exponents and Polynomials
31.
()()()
22
952 95 92
45 18
xy x y xy x xy y
xy xy
+= +
=+
34.
()
22 222
22 2 11
422
6596569
30 54
30 54
xy x y xy x xy y
xy xy
xy xy
++
−= ⋅− ⋅
=−
=−
37.
()()
()
()
22 2 2
223
b a ab b b a b ab b b
ab ab b
−−+=− −−−
=− + −
39.
()( )
() () () ()
22
22
573
7 3 57 53
733515
73815
xyxy
xx xy yx yy
xxyxyy
xxyy
++
=++ +
=++ +
=+ +
Section 5.4 Polynomials in Several Variables
42.
( )( )()()()()()()()()
22
22
3253235 2 5
61525
6135
xyxy xx xy yx yy
xxyxyy
xxyy
− + = + +− +−
=+ −−
=+ −
44.
( )( )()()()()()()()()
22
22
712372 7 312 13
14 21 2 3
14 19 3
xy xy xy xy xy xy
xy xy xy
xy xy
+−= +−+ +−
=−+−
=−−
48.
()()()()
22 2
22
5255
10 25
xy xy xy
xy xy
−= − +
=−+
49.
( ) () ()()()
22 2
22 2 22 2
4224
2
2
xy x xy y
xxyy
+= + +
=+ +
52.
( ) () ()()()
22 2
22 2 22 2
4224
2
2
xy x xy y
xxyy
−= − +
=− +
Chapter 5 Exponents and Polynomials
54.
()()()()
22
22
55 5 25xyxy x y x y+−=−=−
55.
()()()
2222
11 1 1ab ab ab a b+−= −=−
59.
()()()
2
22 22
42 2
33 3
9
ab a ab a ab a
ab a
+−=−
=−
62.
()()()
()
22
22 2
24 2
710710 7 10
49 100
xy y xy y xy y
xy y
−+=−
=−
65.
()
()
22
3xyx xyy+++
()()
2222
32 22 23
32 23
33
33
44
xx xy y yx xy y
xxyxyxyxyy
xxyxyy
=+++++
=+ + + + +
=+ + +
Section 5.4 Polynomials in Several Variables
68.
()
()()()
()( )
222222
2222
32 22 23
32 23
444
44
44
55
xyx xyy xx xyy yx xyy
xx xy y y x xy y
xxyxyxyxyy
xxyxyy
−−+=−+−−+
−++−+−
=− + − + −
=− + −
71.
()( )
242
11xxyx+++
()()
24 2 4 2
6 424 2
6442
11 1
1
21
xxyx xyx
xyxxxyx
xy xy x x
=+++++
=+++++
=++++
74.
()()()
() ()
22 2
22 22 22 44 22
52551025xy xy xy xy xy−= − + = − +
Chapter 5 Exponents and Polynomials
78.
()()()
() ()( ) ( )
2
22
22 2
33 3
23 3
69
xyxy xy
xxxyy
xxyy
++=+
=+ +
=+ +
The area of the shaded region is
22 2
69xxyy++
square units.
81.
()()
()
()
2
33 33
2
22
33
11
1
xy xy
xy
+−
=−
()
()()
() ()
2
22
2
22
22
22 22
44 22
3
9
299
18 81
xy
xy
xy xy
xy xy
=−
=−
=− +
−+
() ()
()
()
2
2
22 2
2
xyz xyz
xyz
xy yzz
=++ −+
=−+
=− + +
Section 5.4 Polynomials in Several Variables
87.
2
124; 10, 16
4
Nxyxyyx y=−+==
2
124
4
Nxyxyy=−+
88. a.
()( ) ()( )
22
22
1
22
1
21327 1327
2
xy xy
π
π
+
=+
89.
00
80, 96 and 2vs t== =
2
00
2
16
16 80 96
stvts
stt
=− + +
=− + +
90.
00
80, 96 and 4vs t== =
2
00
2
16
16 80 96
stvts
stt
=− + +
=− + +
91.
00
80, 96 and 6vs t== =
2
00
2
2
16
16 80 96
stvts
stt
=− + +
=− + +
94. The corresponding point is
()
4,160 .
95. (2,192)
96. The ball strikes the ground after 6 seconds. That is,
()()
16 6.25 80 2.5 96
100 200 96
196
=− + +
=− + +
=
103. does not make sense; Explanations will vary.
Sample explanation: FOIL is used to multiply a
binomial by a binomial.
104. does not make sense; Explanations will vary.
Chapter 5 Exponents and Polynomials
106. false; Changes to make the statement true will vary.
A sample change is: The coefficient of the
32
xy
term is a 1.
109. Area of rectangle:
()()
22
10 8 18 80xyxyx xyy++=++
Area of two corner squares:
22 2
2xx x+=
Area of shaded region:
()
22222
18 80 2 18 80xxyyxxxyy++ −=−++
111. Note that the shed consists of a rectangular solid
with half a cylinder on top.
Radius of cylinder: x
Length of cylinder: y
Volume of cylinder:
2
rl
π
⋅
Volume of half-cylinder:
2
1
2
rl
π
⋅
shed base top
VVV
=+
112. 3; for
2
LW
R
W
+
=
3
23
23
23
33
22
or
33
LW
R
RL L W L
RL W
RL W
R
LRL
WW
+
=
−=+ −
−=
−=
−−
==
113.
()
6.4 10.2 6.4 10.2 3.8−−− =−+ =
238
58
58
55
xx
x
x
=− −
=−
−
=
115.
3
x
=x⋅x⋅xxxx
x
x⋅x⋅
x=
116.
23 6
3
()
125
5
xx
=