Chapter 6
FACTORING POLYNOMIALS
6.1 Common Factoring and
Factoring by Grouping
Exercises
2. A common factor of two or more integers is
an integer that is a factor of each integer.
4. The greatest common factor of two or more
6. 2
2
28 2 2 7 2 7
35 5 7
63 337 3 7
GCF 7
=⋅⋅= ⋅
=⋅
=⋅⋅= ⋅
=
12. 3
2
2
933
623
GCF 3 3
m mmm
mn mmn
mm m
=⋅⋅⋅⋅
=⋅⋅⋅⋅
=⋅ ⋅ =
14.
2
24 2223
32 22222
mn m n
mn m n n
=⋅⋅⋅⋅ ⋅
= ⋅⋅⋅⋅⋅ ⋅⋅
20.
() ()
()
10 15 5 2 5 3
52 3
yy
y
+= +
=+
22.
()
()
()
22
2
30 6 6 5 6 1
65 1
yy
y
−= −
=−
24.
() ()
()
16 8 8 2 8
82
rt r t
rt
−= −
=−
()
23
43
zz
=−
30.
() ()
()
2
12 18 6 2 6 3
62 3
aaaaa
aa
−= −
=−
38.
()
()
()
23 43 23 23 2
23 2
79 7 9
79
ab ab ab ab a
ab a
+= +
=+
40.
()
() ()
()
22
2
5 20 10 5 54 52
542
yy y y
yy
−+= − +
=−+
Chapter 6 Factoring Polynomials
108
52.
() () ()
()
24 3 3
322
322
18 24 30
63 64 65
63 4 5
xy xy xy
xy xy xy y xy x
xy xy y x
−+
=−+
=−+
54.
()()()()
24 4 42nnn n n++ +=+ +
64.
()()()()
()()
323 32 3
32
xx x xx x
xx
−+ − = −− −
=− −
66.
()()()()
()()
949 94 9
94
nn n nn n
nn
−− − = −+ −
=− +
76.
()()
()( )
2
612510
6252
265
ab ac bc c
ab c cb c
bcac
+−−
=+−+
=+ −
11
1
NN
Sa
dN
−−
−
=−
82.
()
1
1
n
n
n
Saar
Sa r
ar
=+
=+
+
88.
()
12
12
11
22
1
2
A
hb hb
A
hb b
=+
=+
()
12
1
22
2
A
hb b
⋅=⋅ +
94. 4
3
14 2 7
21 3 7
GCF 7
x xxxx
xy xyyy
x
=⋅⋅⋅⋅⋅
=⋅⋅⋅⋅⋅
=
Section 6.2 Factoring Trinomials Whose Leading Coefficient Is 1
109
96.
()
1.08 1.08 1.08pq pq+= +
98.
()
1PPrt P rt+=+
100.
()
180 360 180 2nn−= −
Mindstretchers
1. a. Answers may vary.
b.
Reversing the order of the digits so
that
a is the digit in the thousands
place,
b is the digit in the hundreds
place,
c is the digit in the tens place
2.
(
)
212n n nn nn
a b ab ab a b
++
−= −
3. Answers may vary. Possible answers are
6.2 Factoring Trinomials
Whose Leading
Coefficient Is 1
14.
()
3x+
16.
()
7x+
18.
()( )
298 1 8xx x x++=+ +
30.
()()
2412 2 6ss s s−−=+ −
32. 27xx++
Chapter 6 Factoring Polynomials
110
36.
()()
2318 3 6xx x x−−=+ −
44.
()( )
22
20 12 12 20
210
aaaa
aa
+− =− +
=− −
54.
()()
22
30 5 6rrssrsrs−− =+ −
56.
()()
22
12 27 3 9a abbabab++ =+ +
64.
()
()( )
22
2
5 15 10 5 10 15
523
51 3
zzzz
zz
zz
−− = − −
=−−
=+ −
68.
()
32 2
42 42
qq qqqq
−− = −−
74.
()
3232
2
5 10 15 5 15 10
532
bbbbbb
bb b
++ =+ +
=++
80.
(
)
()()
43222
2
42464 4 616
428
tttttt
tt t
+−= +−
=−+
23 22 2 2 2
90.
(
)
()()
43222
2
17 72 17 72
89
ssssss
ss s
−+ = −+
=−−
92.
()()
211 18 2 9nn nn− +=− −
Section 6.3 Factoring Trinomials Whose Leading Coefficient is Not 1
111
96.
()()
211 30 5 6nn nn++=+ +
The factors represent two whole numbers
that differ by 1
102.
()
()( )
2
157 180
18
1360
18
xx
xx
⎛⎞
−+−
⎜⎟
⎝⎠
⎛⎞
=− − +
⎜⎟
⎝⎠
Mindstretchers
1. Answers may vary.
a. Possible answers: 16, 17−
6.3 Factoring Trinomials
Whose Leading
10.
()
2x−
12.
()
32x−
14.
()()
2
215721 7xx xx++=+ +
16.
()()
2
310737 1yy yy−+=− −
18.
()()
2
211929 1xx xx++=+ +
34.
()
()()
5143 5143
51 3
bb bb
bb
−−+=− +−
=− − +
36.
()()
2
65253525xx x x+−= − +
Chapter 6 Factoring Polynomials
112
38. 2
22714yy++ Prime polynomial
46.
()
()()
22
924153385
33 5 1
yy yy
yy
−+= −+
=−−
52.
()
()()
32 2
6452132157
32 1 7
xxxxxx
xx x
++= ++
=++
60.
()()
22
12 25 12 4 3 3 4aabbabab−+=− −
62.
()()
22
6123423sst t stst−− = + −
64.
()()
22
38 5 35mmnn mnmn−+=− −
70.
()
22
22
42648
2 2 13 24
aabb
aabb
+−
=+−
()()
45 4 3
pp q pq
=+−
74.
()
32 23 4
22 2
2
24 6 18
64 3
xy xy xy
xy x xy y
−−+
=− + −
80.
()
()( )
22
836204295
4521
xx xx
xx
+−= +−
=+ −
88.
()
1x−
90.
()
()()
32 2
32 321
31 1
wwwwww
ww w
−−= −−
=+−
92. a.
()
2222
24rrrr
ππ
−=−
Section 6.4 Factoring Perfect Square Trinomials, the Difference of Squares, and the Sum or Difference of
Cubes
113
Mindstretchers
1. Answers may vary. Possible answers:
a. 7; 11
2.
6.4 Factoring Perfect Square
Trinomials, the Difference
of Squares, and the Sum
or Difference of Cubes
Exercises
8. Neither
10. Perfect square trinomial
12. Difference of squares
24.
()
2
214 49 7yy y−+=−
26.
()
2
221 1xx x++=+
30. 210 25yy−− Prime polynomial
32.
()
2
2
25 20 4 5 2bb b−+=−
34.
()
2
2
9 241634yy y++=+
42.
()
2
42 2
44 2xx x++=+
44.
()
()
22
2
12 24 12 12 2 1
12 1
yy yy
y
++= ++
=+
50.
()()
2111nnn−= + −
52.
()()
216 4 4xxx−=+ −
54.
()()
2
225 15 15ttt−= + −
Chapter 6 Factoring Polynomials
114
62.
()()
22
49 7 7cd cd cd−= + −
70.
(
)
()()
32
50 18 2 25 9
25353
xy x y xy x
xy x x
−= −
=+−
72.
()
()()
42 42
22
9819 9
93 3
xy xy
xy xy
−= −
=+ −
78.
()()()
(
)
()()()
22
1
11
yab ab aby
aby y
−−−= − −
=− + −
80.
()()()
()
()()()
22
99
33
yc x yc yc x
yc x x
−− −= − −
=− + −
88. 22
pq+ Prime polynomial
92.
333
22
27 3
xx
+=+
96. 4
8n+ Prime polynomial
98.
()
()
()
33
2
3243 8
32 24
xx
xxx
−= −
=− ++
100.
()
36 3 3
128 2 2 64
nn n n
+= +
106.
()
()
2
85698242849
22 7
tttttt
tt
−+= −+
=−
108.
()
()()
32
12 27 3 4 9
32323
xy xy xy y
xy y y
−=−
=+−
116. Perfect square trinomial
118. Neither
Section 6.5 Solving Quadratic Equations by Factoring
115
122.
()
()()
22 2
25 in , or 5 5 inxxx−+−
124.
()
()
2
2
2
16,000 32,000 16,000
16,000 1 2
16,000 1
rr
rr
r
++
=++
=+
Mindstretchers
1.
()()
3599 3600 1
60 1 60 1
=−
=− +
59 and 61
2. a.
x
b.
2
x
3. The length of each side of the large outer
square is .ab+ The area of the large outer
square therefore is
()
2.ab+ The area of the
6.5 Solving Quadratic
Equations by Factoring
Exercises
2. The zero-product property states that if the
10. Quadratic
12.
()()
210xx−−=
20
2
x
x
−=
=
or 10
1
x
x
−=
=
14.
()
72 3 0
230
23
t
t
t
−+=
+=
=−
20.
()()
54 54 0xx−+=
54 0
45
x
x
−=
−=−
or 54 0
45
x
x
+=
=−
Chapter 6 Factoring Polynomials
116
24.
()
230
30
yy
yy
+=
+=
0y= or 30
3
y
y
+=
=−
30.
()( )
290 0
9100
yy
yy
−− =
+−=
90
9
y
y
+=
=−
or 10 0
10
y
y
−=
=
2
2
36.
()()
2
025 10 1
05151
yy
yy
=++
=+ +
510
51
1
5
y
y
y
+=
=−
=−
40.
()
2
041x=+
410
41
1
x
x
x
+=
=−
=−
31
1
3
n
n
=
=
5
n
=
46.
2
2
310
rr
+=
()()
360
tt
+−=
30
3
t
t
+=
=−
or 60
6
t
t
−=
=
52.
()()
2
2
317 10
317100
32 50
kk
kk
kk
+=−
++=
++=
Section 6.5 Solving Quadratic Equations by Factoring
117
56.
()
2
2
36
360
320
mm
mm
mm
=
−=
−=
30
0
m
m
=
=
or 20
2
m
m
−=
=
3
3
60.
()
()()
2
2
2
12 48
12 48 0
12 4 0
220
x
x
x
xx
=
−=
−=
+−=
20
2
x
x
+=
=−
or 20
2
x
x
−=
=
68.
()()
()( )
2
2
6110
5610
540
140
mm
mm
mm
mm
−+=−
−−=−
−+=
−−=
10
1
m
m
−=
=
or 40
4
m
m
−=
=
72.
()()
()()
2
2
35 116
32516
31450
31 50
xx x
xx x
xx
xx
+−=
+−=
−−=
+−=
310
31
1
x
x
x
+=
=−
=−
or 50
5
x
x
−=
=
Chapter 6 Factoring Polynomials
118
80.
()
2
2
927
9270
930
bb
bb
bb
=
−=
−=
90
0
b
b
=
=
or 30
3
b
b
−=
=
84.
()()
73 73 0mm−+=
73 0
37
7
3
m
m
m
−=
−=−
=
or 73 0
37
7
3
m
m
m
+=
=−
=−
86. Quadratic
88.
()
()
11 325
2
1
2 1 2 325
nn
nn
−=
⋅−=⋅
90.
222
2
2
12 13
144 169
25 0
x
x
x
+=
+=
−=
3
2
t
=
The diver will hit the water in 3,
2 or 1.5 sec.
94.
()()
2
2 8 2 2 10 40
xxx
++ =
A border 1 in. wide.
96. a.
()
()()
2
0.04677 4
0.04677 2 2
r
rr
π
π
−
=+−
b.
()()
0.04677 2 2 0rr
π
+−=
20
2
r
r
+=
=−
or 20
2
r
r
−=
=
2 cm
Mindstretchers
20
2
x
x
+=
=−
or 50
5
x
x
−=
=
b.
x-intercepts:
() ()
2, 0 and 5, 0−