Chapter 6
Factoring Polynomials
6.1 Check Points
1. a.
32
22
18 3 6
15 3 5
xxx
xx
=⋅
=⋅
The GCF is
2
3.x
32 2
22
xy xyxy
xy xy
=⋅
=
The GCF is
2
.xy
5.
32 2 2
2
8142242721
2(4 7 1)
x y x y xy xy x y xy x xy
xy x y x
−+=⋅−⋅+⋅
=−+
6.
45 34 5 2 33 2 22 2
233 22
16 24 20 4 4 4 ( 6 ) 4 5
4(4 6 5)
a b a b ab ab a b ab a b ab
ab a b a b
−+ −=−⋅−⋅−−⋅
=− − +
Chapter 6 Factoring Polynomials
6.1 Concept and Vocabulary Check
1. factoring
6.1 Exercise Set
1. The GCF of 4 and 8x is 4.
2. The GCF of 5 and 15x is 5.
3. The GCF of
2
12x and 8x is 4.x
8. The GCF of
5
10y,
2
20 ,y and 5y− is 5.y
9. The GCF of
2
,xy xy , and
3
xy is .xy
10. The GCF of
23
,3xy xy, and
2
6x is
2
.x
15.
()
444 41
41
yy
y
−=⋅−⋅
=−
16.
()
555 1yy−= −
19.
()
30 12 6 5 6 2
65 2
xx
x
−=⋅−⋅
=−
22.
()
2
66xxxx+= +
23.
()
22
2
18 12 6 3 6 2
63 2
yy
y
+=⋅ +⋅
=+
24.
()
22
20 15 5 4 3yy+= +
27.
()
2
13 25 13 25
13 25
yyyyy
yy
−=⋅−⋅
=−
28.
()
2
11 30 11 30yyyy−= −
29.
464 42
42
9279193
yyy yy
+=⋅+⋅
32.
()
12 4 4 3xx x x−= −
33.
()
() ()
()
22
2
12 16 8 4 3 4 4 4 2
43 4 2
yy y y
yy
+−= + −
=+−
Section 6.1 The Greatest Common Factor and Factoring by Grouping
35.
()
() ()
()
432
22 2 2
22
9186
33 36 32
33 62
xxx
xx xx x
xx x
++
=++
=++
38.
()
53 2 23
26 13 39 13 2 3yyy yyy−+ = −+
39.
23
10 20 5xxx−+
() ( )
()
()
2
2
52 54 5
524
xxxxx
xxx
=− +
=−+
43.
()
()
()
32 2
2
6932 33
32 3
x y xy xy x y xy
xy x y
+= ++
=+
46.
23 2 2
27 18 45xy xy xy−+
()
() ()
()
2
2
93 92 95
93 25
xy xy xy y xy x
xy xy y x
=−+
=−+
47.
32 3 2
32 24 16xy xy xy−−
() () ()
()
222
2
84 8382
8432
xy xy xy x xy
xy xy x
=−−
=−−
51.
432 22
83216 8(42)xxx xxx−+ + =− −−
52.
432 22
1896 3(632)xxx xxx−++=− −−
53.
32 2
462(23)a b ab ab a b−+=− −
54.
23 2
9123(34)a b ab ab ab−+=− −
59.
()()()()
24 2 2 4xx x x x+− +=+ −
60.
()()()()
38 3 3 8xx x x x+− += + −
64.
()()()()
()( )
771
71
xxy xy xxy xy
xy x
+−+= +− +
=+ −
65.
()
43 1 3 1xx x++ +
()()
()()
43 113 1
3141
xx x
xx
=+++
=+ +
Chapter 6 Factoring Polynomials
68.
()
2
97272xx x++ +
()()
()
()
2
2
972172
729 1
xx x
xx
=+++
=+ +
71.
()
()
()()
()()
22
3515 3 515
35 3
35
xxx xx x
xx x
xx
+−−= + +−−
=+−+
=+ −
74.
()
()
()()
()
()
32 32
2
2
3412 3 412
34 3
34
xxx xx x
xx x
xx
−+−=− +−
=−+−
=− +
78.
()()
()()
()()
6212 6 212
62 6
62
xy x y xy x y
xy y
yx
+++=+++
=+++
=+ +
()()
()( )
3252
235
xx y yx y
xyxy
=−+−
=− +
82.
22
10 12 35 42xxyxyy−+−
()( )
22
10 12 35 42
xxy xyy
=−+−
()
()
2
41 3
xx
=− −
85.
()()
()()
2
xaxbxabxxabxa
xaxb
−−+ = −− −
=− −
()()
Section 6.1 The Greatest Common Factor and Factoring by Grouping
87.
333 22 2 2
24 30 18
xyz xyz xyz++
()
() ()
()
2222 2
222
64 6563
64 53
xyz xyz xyz y xyz z
xyz xyz y z
=++
=++
90.
()()
()
()
33 3 3
3
52 10 1 5 2 5
512
xxyyx yx
xy
−+ − = − + −
=− +
93.
()()
()
() ()()
()
()
5 432
54 32
42
42
33 55
33 55
31 151
13 5
xxxxx
xx xx x
xx xx x
xxx
−+−+−
=−+−+−
=−+−+−
=− ++
94
5432
77 33
xxxxx
−+−+−
96. The area of the square is
2
44 16.xx x⋅= The area
of each circle is
2
x
π
. The area of both circles is
2
2.x
π
So the shaded area is the area of the square
minus the area of the two circles, which is
222
b.
()
2
64 16 16 4xx xx−= −
c. Substitute 3 for x in the factored polynomial.
() ()
16 3 4 3 48 1 48⋅−= =
b.
()
2
72 16 8 9 2xx x x−=−
c. Substitute 4 for x in the factored polynomial.
()()()
849 24 329 8 321 32⋅−⋅= −= =
You do get the same answer as in part (a) but
this does not prove your factorization is correct.
99. Use the formula for the area of a
Chapter 6 Factoring Polynomials
100. Use the formula for the area of a
rectangle,
A
lw=⋅
. Substitute
4
714
xx−
for A
and 7
x
for w.
A
101. – 106. Answers will vary.
107. makes sense
108. does not make sense; Explanations will vary.
Sample explanation: You can always check your
factoring by multiplying.
112. true
113. false; Changes to make the statement true will vary.
A sample change is:
22
ab+
is not factorable.
117. Answers will vary. One example is
2
51048.xxx+−−
118. The graphs do not coincide.
Factor out the GCF from the left side.
()
36 3 2.xx−−=− +
Change the expression on the
120. The graphs do not coincide.
Factor by grouping.
()()
()()
2
22 212
21
xxx xx x
xx
+++= ++ +
=+ +
Change the expression on the right side to
()()
21.xx++
x
5
5
−5
(−3, −2)
Section 6.2 Factoring Trinomials Whose Leading Coefficient is 1
123. First, find the slope
()
52 3 1
473
m−
===
−−−
Write the point-slope equation using
()()
11
1 and , 7, 2 .
mxy
==−
124.
24 8and24 6×= +=
6.2 Check Points
1.
2
56xx++
Factors of 6 6,1 6, 1 2,3 2, 3
Sum of Factors 7 7 5 5
−− −−
−−
2.
2
68xx−+
Factors of 8 8,1 8, 1 2, 4 2, 4
Sum of Factors 9 9 6 6
−− −−
−−
3.
2
310xx+−
Factors of 10 10,1 10, 1 5, 2 5, 2
Sum of Factors 9 9 3 3
−− −− −
−−
7. First factor out the common factor of
2.x
32 2
2( 4)( 7)
xx x
=−+
8. First factor out the common factor of 2.−
22
210282( 514)yy yy−−+=− +−
Continue by factoring the trinomial.
22
210282( 514)
2( 2)( 7)
yy yy
yy
−−+=− +−
=− − +
2. completely
3. +10
4. 6
Chapter 6 Factoring Polynomials
6.2 Exercise Set
1.
2
76xx++
Factors of 6 6,1 6, 1 3, 2 3, 2
Sum of Factors 7 7 5 5
−− −−
−−
76
xx
=++
2.
2
98xx++
Factors of 8 8,1 8, 1
Sum of Factors 9 9
−−
−
3.
()()
()
2
710 5 2
52 10; 5 2 7
xx x x++=+ +
=+=
4.
()()
()()
2
914 7 2
27 14; 27 9
xx x x++=+ +
=+=
7.
()()
()
2
712 4 3
43 12; 4 3 7
xx x x−+=− −
−−= −+−=−
11.
()()
()
2
815 5 3
53 15; 5 3 8
yy y y−+=− −
−−= −+−=−
12.
()()
2
87 7 1
yy y y−+=− −
14.
()()
()( )
2
328 7 4
74 28; 743
xx x x+−=+ −
−=− +−=
15.
()()
2
10 39 13 3
yy y y+ −=+ −
17.
()()
()()
215 5 3
53 15; 53 2
xx x x−−=− +
−=−−+=−
18.
()()
()()
2
45 5 1
51 5; 51 4
xx x x−−=− +
−=−−+=−
19.
()()
2
28 4 2
xx x x−−=− +
integers whose product is 12 and whose sum is 4.
22.
2
45xx++ is prime because there is no pair of
integers whose product is 5 and whose sum is 4.
Section 6.2 Factoring Trinomials Whose Leading Coefficient is 1
26.
2
410xx+− is prime because there is no pair of
integers whose product is −10 and whose sum is 4.
29.
()( )
()( )
2
18 65 5 13
51365; 51318
yy yy−+=− −
−− = −+−=−
33.
2
75yy−+ is prime because there is no pair of
integers whose product is 5 and whose sum is −7.
34.
2
15 5yy−+ is prime because there is no pair of
integers whose product is 5 and whose sum is −15.
38.
()()
()()
22
914 2 7
2714; 27 9
xxyyxyxy−+ =− −
−−= −+−=−
41.
()()
()()
22
18 45 15 3
15 3 45; 15 3 18
aabbabab−+ =− −
−−= −+−=−
binomial.
()
()()
22
315183 56
32 3
xx xx
xx
++= ++
=+ +
46.
()
()()
22
33183 6
33 2
yy yy
yy
+−= +−
=+ −
47.
()
22
10 40 600 10 4 60
xx xx
−−= −−
50.
()
()()
214242 712
23 4
xx xx
xx
−+= −+
=− −
Chapter 6 Factoring Polynomials
53.
()
()()
32 2
412 724 318
463
xxxxxx
xx x
+−= +−
=+−
56.
()
()()
32 2
39543 318
363
rr rrrr
rr r
−−= −−
=−+
59.
()
()()
43 222
2
310 310
52
xx xxxx
xx x
−− = −−
=−+
60.
()
()()
43 222
2
22 120 22 120
12 10
xx xxxx
xx x
−+ = −+
=− −
63.
()
()()
22
15 45 60 15 3 4
15 4 1
xy xy x x y y
xy y
+−= +−
=+−
()()
66.
()
()()
3223 2 2
23 23
3
x y x y xy xy x xy y
xy x y x y
−−=−−
=− +
69.
550455(109)
5( 9)( 1)
xx xx
xx
−+ −=− −+
=− − −
70.
22
336333(1211)
xx xx
−+ −=− −+
73.
32 2
268 2(34)
2( 4)( 1)
xxx xxx
xx x
−− +=− +−
=−+−
74.
32 2
3624 3(28)
3( 4)( 2)
xx x xxx
xx x
−+ + =− −−
=− − +
77.
()() ()
()
()
2
2
20
20
abx abx ab
abx x
+++−+
=+ +−
Section 6.2 Factoring Trinomials Whose Leading Coefficient is 1
79.
()()
()
2
0.5 0.06 0.2 0.3
0.2 0.3 0.06; 0.2 0.3 0.5
xx x x++=+ +
=+=
80.
()()
()
2
0.5 0.06 0.6 0.1
0.6 0.1 0.06; 0.6 0.1 0.5
xx x x−−=− +
−=−+=−
83. a.
()
()()
22
16 16 32 16 2
16 2 1
tt tt
tt
−++=− −−
=− − +
b. Substitute 2 for
t
in the original polynomial:
() ()
()
2
2
16 16 32 16 2 16 2 32
16 4 32 32
64 64
0
tt−++=− + +
=− + +
=− +
=
b. Substitute 3 for t in the original polynomial:
() ()
()
2
2
16 32 48 16 3 32 3 48
16 9 96 48
144 144
0
tt−++=− + +
=− + +
=− +
=
Substitute 3 for t in the factored polynomial:
()() ( )()
()()
16 3 1 16 3 3 3 1
16 0 4
0
tt−−+=− − +
=−
=
The answers are the same.
This answer means that after 3 seconds you hit
91. does not make sense; Explanations will vary.
Sample explanation:
2
1xx++ is prime.
92. makes sense
93. false; Changes to make the statement true will vary.
A sample change is:
2
20xx++ is prime.
94. false; Changes to make the statement true will vary.
A sample change is: Some trinomials have two
identical factors. For example
98. In order for
24
xxb
++
to be factorable, b must be
an integer with two positive factors whose sum is 4.
The only such pairs are 3 and 1, or 2 and 2.
()()
()()
2
2
31 43
22 44
xx xx
xx xx
++=++
++=++
Therefore, the possible values of b are 3 and 4.
Chapter 6 Factoring Polynomials
100.
220 99
nn
xx
++
Notice that
()
2
2.
nn
xx
=
102.
()
()()
32 2
428484 712
443
xxxxxx
xx x
−+= −+
=−−
The box has the following dimensions:
length =
()
82 24
xx
−= −
103. The graphs coincide.
This verifies the factorization
()()
2
56 2 3.
xx x x
−+=− −
104. The graphs do not coincide.
106. The graphs do not coincide.
()
22
2862 43
xx xx
++= ++
108. 65 30xy−=
Section 6.3 Factoring Trinomials Whose Leading Coefficient is Not 1
6.3 Check Points
1. Factor
2
5148xx−+
by trial and error.
Step 1
2
5 14 8 (5 )( )xx xx−+=
Step 2 The number 8 has pairs of factors that are
either both positive or both negative. Because the
middle term,
14x−
, is negative, both factors must
be negative.
Step 3
2
required middle term
Possible Factors of Sum of Outside and
Inside Products
5148
(5 4)( 2) 10 4 14
xx
xx xx x
−+
−− −−=−
2. Factor
2
6197xx+−
by trial and error.
Step 1 Find two First terms whose product is
2
6.x
Step 3
2
Possible Factors of Sum of Outside and
Inside Products
6197
(6 1)( 7) 42 41
(6 7)( 1) 6 7
(6 1)( 7) 42 41
xx
xx xx x
xx xxx
xx xxx
+−
+− −+=−
−+ −=−
−+ −=
3. Factor
22
3134xxyy−+
by trial and error.
Step 1 Find two First terms whose product is
2
2.x
22
3 13 4 (3 )( )xxyyxx−+=
Step 2 The last term,
2
4,y
has pairs of factors that
are either both positive or both negative. Because
the middle term, 13 ,xy− is negative, both factors
must be negative. Thus the last term has possible
factorizations of
2(2)or (4).yy yy−− −−
Step 3
22
Possible Factors of Sum of Outside and
Inside Products
3134
xxyy
−+
4. Factor
2
310xx−−
by grouping.
3 and 10, so 3( 10) 30.ac ac==− =−=−
(3 5)( 2)
xx
=+ −
5. Factor
2
8103xx−+
by grouping.
()
8 and 3, so 8 3 24.acac== ==
The factors of 24 whose sum is –10 are –6 and –4.
22
81038463
4(2 1) 3(2 1)
xx xxx
xx x
−+= −−+
=−−−
Chapter 6 Factoring Polynomials
6.3 Concept and Vocabulary Check
1. greatest common factor
2. 3
6.3 Exercise Set
1. Factor
2
253 xx++
by trial and error.
Step 1
()()
2
2 5 3 2 xx xx++=
Step 2 The number 3 has pairs of factors that are
either both positive or both negative. Because the
middle term,
5x
, is positive, both factors must be
positive. The only positive factorization is
()( )
13.
2. Factor
2
352xx++ by trial and error.
Step 1
()()
2
3 5 2 3 xx xx++=
Step 2 The number 2 has pairs of factors that are
either both positive or both negative. Because the
middle term, 5x, is positive, both factors must be
positive. The only positive factorization is
()( )
12.
Step 3
3. Factor
2
3134xx++ by trial and error.
The only possibility for the first terms is
()()
2
33.xx x=
Because the middle term is positive and the last
Thus,
()()
313431 4xx xx++=+ +.
4. Factor
2
273xx++ by trial and error.
The only possibility for the first terms is
()()
2
22.xx x=
Because the middle term is positive and the last
term is also positive, the only possible factorization
of 3 is
()( )
13 .
()()
()( )
2434
42 3
xx x
xx
=+++
=+ +
6. Factor
2
21935xx++ by grouping.
()
2 and 35, so 2 35 70.ac ac== ==
The factors of 70 whose sum is 19 are 14 and 5.
Section 6.3 Factoring Trinomials Whose Leading Coefficient is Not 1
7. Factor
2
5163yy−+ by trial and error. The first
terms must be 5 and .yy Because the middle term is
negative, the factors of 3 must be −3 and −1.
8. Factor
2
5176yy−+ by trial and error. The first
terms must be 5 and .yy Because the middle term is
negative, the factors of 6 must be −3 and −2 or
1−
and
6.−
()()
2
53 25 136
yy yy
−−=−+
9. Factor
2
34yy+−
by trial and error.
()()
()()
2
2
31 43 114
31 43 114
yy y y
yy y y
+−=−−
−+=+−
10. Factor
2
34yy−−
by grouping.
3 and 4, so 12.ac ac==− =−
11. Factor
2
31310xx+−
by grouping.
3 and 10, so 30.ac ac==− =−
12. Factor
2
3145xx+−
by grouping.
3 and 5, so 15.ac ac==−=−
The factors of
15−
whose sum is 14 are 15 and
1.−
()()
()()
2
2
37 13 107
31 7 3 22 7
xx xx
xx x x
−−=−+
− −=−+
Thus,
()()
2
322731 7xx xx−+=− −
.
14. Factor
2
3107xx−+
by trial and error.
2
31 7 3 22 7
xx x x
− −=−+
()()
()()
2
53 15 83
51 35 16 3
yy yy
yy y y
−−=−+
−−=−+
Thus,
()()
2
516351 3.yy yy−+=− −
17. Factor
2
31710xx−+
by grouping.
3 and 10, so 30.ac ac== =
The factors of 30 whose sum is −17 are −15 and −2.
3 and 28, so 84.ac ac==− =−
The factors of −84 whose sum is −25 are 3 and −28.
Chapter 6 Factoring Polynomials
19. Factor
2
6114ww−+
by grouping.
6 and 4, so 24.acac== =
20. Factor
2
61712ww−+
by grouping.
21. Factor
2
8334xx++
by grouping.
8 and 4, so 32.acac== =
22. Factor
2
7436xx++
by grouping.
7 and 6, so 42.acac== =
23. Factor
2
53314xx+−
by trial and error.
()()
2
57 25 314
xx xx
−+=+−
24. Factor
2
32216xx+−
by trial and error.
()()
2
38 23 216
xx xx
+−=+−
25. Factor
2
14 15 9yy+− by trial and error. The sign in
one factor must be positive and the other negative.
Thus,
()()
2
14 15 9 7 3 2 3 .yy y y+−=− +
()()
()()
2
2
38236 724
38236 724
yy yy
yy yx
+−=+−
−+=−−
()()
()()
()( )
2
2
2
61 36 193
63 16 93
312 3 6 113
xx x x
xx xx
xx xx
−−=−+
−−=−+
−−=−+
so
673xx−+ is prime.
28. Factor
2
932xx++ by grouping.
9 and 2, so 18.acac== =
()( )
()()
2
2
5159 25 509
535325 309
zz zz
zz zz
−−=−+
−−=−+
Section 6.3 Factoring Trinomials Whose Leading Coefficient is Not 1
30.
2
9124zz++
Use trial and error until the correct factorization is
obtained. The signs in both factors must be positive.
31. Factor
2
15 2yy−− by grouping.
15 and 2, so 30.acac==−=−
32. Factor
2
15 13 2yy+− by grouping.
33. Factor
2
529xx++ by trial and error. The signs in
both factors must be positive.
34.
2
351xx−+
Use trial and error. The signs in both factors must
35. Factor
2
10 43 9yy+− by grouping.
36.
2
16 46 15
yy
−+
Use trial and error. The signs of both factors must
be negative. Try various combinations until the one
()()
()()
2
2
85238 3415
83258 4615
yy yy
yy yy
−−=−+
− −=−+
2
()()
()()
2
2
41218 21
41218 21
xx xx
xx xx
−+=+−
+−=−−
()()
22
822582205
24 1 54 1
xx xxx
xx x
−+=−−+
=−−−
()( )
3132
yy
=− −
40.
2
954
yy
+−
()()
2
2
34319 94
yy yy
−+=−−
Chapter 6 Factoring Polynomials
41. Factor
2
20 27 8
xx
+− by grouping.
20 and 8, so 160.acac==−=−
42. Factor
2
15 19 6
xx
−+ by grouping.
15 and 6, so 90.acac===
43.
()()
22
23 2
x xyy xyxy
++=+ +
44.
()()
22
34 3
xxyy xyxy
++=+ +
46. Factor
22
3116
xxyy
++ by trial and error.
47. Factor
22
299
xxyy
−+ by trial and error until the
correct factorization is obtained. The signs in both
factors must be negative.
48. Factor
22
352
xxyy
+− by grouping.
3 and 2, so 6.ac ac==−=−
49. Factor
22
656
xxyy
−− by grouping.
()()
2332
xyxy
=− +
22
()()
()()
23 5 3 5
352
xx y yx y
xyxy
=−+−
=− +
51. Factor
22
15 11 14xxyy+− by grouping.
()()
35 7 25 7
xx y yx y
=+−+
combinations until the one with the correct middle
term is found.
()( )
22
5 3 10 15 53 10
xy x y x xy y
−−=−+
()()
2
252115
aba b a ab b
2
++=++
Thus,
()()
22
275 25aabb abab++=+ +.
Section 6.3 Factoring Trinomials Whose Leading Coefficient is Not 1
54. Factor
22
252aabb++ by grouping.
55. Factor
22
15 6aabb−− by grouping.
15 and 6, so 90.acac==−=−
56. Factor
22
314aab b−− by trial and error. The sign
must be positive in one factor and negative in the
57. Factor
22
12 25 12xxyy−+ by grouping.
12 and 12, so 144.acac== =
58. Factor
22
12 7 12xxyy+− by grouping.
12 and 12, so 144.ac ac==− =−
59.
2
42630xx++
60.
2
41810xx−−
First factor out the GCF, 2. Then factor the resulting
61.
9624xx−−
First factor out the GCF, 3. Then factor the resulting
trinomial by trial and error or grouping.
trinomial by trial and error or grouping.
()
22
12 33 21 3 4 11 7
xx xx
−+= −+
()( )
62 1 3 2
yy
=−+
()()
42 3 2 1
yy
=− +
67.
32
34xxx++
68.
314 8xxx++
First factor out the GCF, x. Then factor the resulting
trinomial by trial and error or grouping.
Chapter 6 Factoring Polynomials
71.
32
93912yyy−+
First factor out the GCF, 3y. Then factor the
resulting trinomial by trial and error or grouping.
()
()()
32 2
9391233134
33 1 4
yyyyyy
yy y
−+= −+
=−−
74.
()
()( )
32 2
80 80 60 20 4 4 3
20 2 1 2 3
zzzzzz
zz z
+−= +−
=−+
77.
()
()()
54332
3
10 17 3 10 17 3
2351
xxxxxx
xx x
−+= −+
=−−
78.
()
54332
15 2 15 2 1
xxxxxx
−−= −−
83.
()
()()
22
8 3484241742
24 7 6
xy xy y y x x
yx x
+−= +−
=−+
84.
()
()()
22
6 2 60 2 3 30
23 10 3
xy xy y y x x
yx x
−− = −−
=−+
87.
24 4 4
42
32 20 12
4(8 5 3)
xy xy y
yx x
−++
=− − −
()
()
()( )()
2
10 1 3 2
10 1 3 2 1
yxx
yxx
=+ +−
=+ −+
90.
() () ()
2
61 331151
yx yx y
++ ++ +