FACTORING POLYNOMIALS
1) First determine if a common monomial factor (Greatest Common Factor) exists. Factor trees may be used to find the
GCF of difficult numbers. Be aware of opposites: Ex. (a-b) and (b-a) These may become the same by factoring -1
from one of them.
3 12 3 4
3 3
6 6 2) If the problem to be factored is a binomial, see if it fits one of the following situations.
A. Difference of two squares:
B. Sum of two squares:
9 25 3 5 3 5
25 5 5 C. Sum of two cubes:
8
5 5
does not factor (it is prime).
27 2
3 4 6
9
Note: Resulting trinomial does not factor.
D. Difference of two cubes:
64 4 4
16
Note: Resulting trinomial does not factor.
E. If none of these occur, the binomial does not factor.
3) If the problem is a trinomial, check for one of the following possibilities.
A. Square of a binomial:
2 6 9 3 3 3
4 20
25 2 5
B. If 1, use reverse foil or trial and error method:
7 12 3 4
7 12 3 4
3 18 6 3
3 18 6 3
C. If 1, use trial and error method. (Grouping may also be used.)
4) If factoring a polynomial with four terms, possible choices are below.
A. Group first two terms together and last two terms together.
5 5 5 5 5 5
2 6 3
3
2 6 3
2 3 3 2
B. Group first three terms together.
6
9
6
C. Group last three terms together.
6 9 9 6
9 3
3
3
3 3 3 BE SURE YOUR ANSWERS WILL NOT FACTOR FURTHER!
All answers may be checked by multiplication.
3
3
3 3
PRACTICE PROBLEMS:
9
2. 5
1.
15 3. 12 20 21. 3
10
22.
5 84
23. 8 16 28
8 16
4. 36
24. 12 31
5. 25 25. 6 7 10
6. 4 49
26. 8 6 9
7. 27. 6 100
8
31. 12. 5 40
20
2 2
30. 11. 27
13. 2
29.
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2
28. 2 14 8. 9 9.
20
14. 64
15. 10
25
16. 4
16
17. 16
56
18. 20
16
49
4
19. 9
20. 2
16
8 8
32. 128
25
3 75
33. 16 2 34. 2 6
35. 6
23
20
36. 9
15
4
3
37. 8! 6! 9
25
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39. 16 " 20
40. 32
ANSWERS:
1.
9
5. 5
5 8. 3
3
9
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27. 3 2 2 1
8 38. 5 5 4
2
4
10.
2
35. 3
or 5
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4
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1
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4
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4
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5
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2
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2 4
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1 1
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3. 43 5 3
6. 2
3 11. 3 14. 2. 5
16
7
12 7
3 2 3
30. 1 1
4 37. 4!
3 2! 3
MORE PRACTICE PROBLEMS:
41. 6 16
61. 4
42. 10
62. 15
43. 24
1 4
12
29
3
2
63. 8# 2# 3
44. 3
2
64. 35
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8
30
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3
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20
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6
68. 25$ 15$ 18
49. 10
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42
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12
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13 4
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75. 50
10 12
56. 2 2 4
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30
57. 2 4 6
77. 18
20
58. 3 6 9
78. 3 22
59. 5 35
60. 3 50
18
18
7
79. 15 18 24
80. 4 25
21
6
ANSWERS:
41. 8 2
42. 6 4
45. 6 5
46. 8 1
47. 49. 2
50. 9 8
51. 9 8 53. 7 6
54. 57. 2 3 58. 3 3 2 1
65. 5
2%
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1
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2
6 3
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43. 2 1
1 4# 3
3
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78. 3 1 7
73. 2 5 3 2
76. 52
79. 35
1
8 6
56. 2 2 1
64. 5
71. 2
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3 3 2
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52. 60. 3 1
7 1
61.
3 5$ 6
69.
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70. '#(!) *++, ) %-,#).
3
48. 3 2
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1
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44. 2 1
1
55. 4 3 2
1 4
2 3 2
68. 5$
5
4 5 1
77. 23 5 3
80. 4 1 6
2
MORE PRACTICE PROBLEMS:
81. 125 1
101.
5 2
82. " 64
102.
343
103.
40
84. 8 48
104.
3 5-
85. 7
105.
8- 125. 106.
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6
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36
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110.
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81
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5
12
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9
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8 118.
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6
0
120.
14
96.
97. 0.04"
98. 4 99. 64
0.28"
40
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64
0
100.
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10
21
48
49
ANSWERS:
81. 5 1 25
5
85. 4 3
86. '#(!) *++, ) %-,#).
88. 6
89. 9 2
1
92. 25
1 25 5
95. 8 5 3
1
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12
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49
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112. 6 36
3
119. 6
82. "
1
6
5 3 5
2 2
1
90. 8 5
10- .
25.
3
91. 3 4
& &
2 3 94.
0.7
2 98. 4
104. 3 5-
106. 9 &
9
4
8
120. 7
107. 5
3
111. 7
114. 7 2
115.
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.
110. 14 113. 9
117. 7
87. 3 2
4
3 12 3 101. '#(!) *++, ) %-,#).
3 5 4
109. 4 84. 12 0
100. 12
103. 42
83. 6
97. 0.2"
16
0
9
4
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3 7
118. '#(!) *++, ) %-,#).
116.
121.
3
122.
34 24
8
16
121
123.
131.
0.6
132.
4 13 35
133.
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134.
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135.
40
136.
15" 15" 90
137.
0.04 0.49
124.
42
125.
9 24
126.
127.
10"
128.
16
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138.
129.
27 30 8
139.
2 54
130.
1
140.
16
0
29" 21
0
0.09
7 3
7
18
5
25
ANSWERS:
121. 12 3
123. 2
11 11
127. 5" 3 2"
130. 7
122. 4
124. 7 6
128. 2
1 1 1 2 " 3
139. 2 3
3
9
129. 9
2 3 4
0
140. 1 53
132. 5 4
135. 8
0.7 0.2 0.7
0
126. 1 3 1
131. 0.3
1
134. 7 1
137. 0.2
0
125. 3 4
7 8 1
133. '#(!) *++, ) %-,#).
136. 15"
138. 7
3 5 1
2 9
0
3