1) First determine if a common monomial factor (Greatest Common Factor) exists Factor trees BE SURE YOUR ANSWERS WILL NOT FACTOR FURTHER
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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.3ݱ ൣ 12 ൩ 3
ݱ ൣ 4ቘ ݲൣ 3ݱݲ൩ ݱݲݱ ൣ 3ቘ 62) If the problem to be factored is a binomial, see if it fits one of the following situations.
A. Difference of two squares:
9ݱ ൣ 25ݲ൩3ݱ ൢ 5ݲቘ3ݱ ൣ 5ݲቘB. Sum of two squares:
ൢ ݛ does not factor (it is prime).C. Sum of two cubes:
8ݱൢ 27ݲ൩2ݱ ൢ 3ݲቘ4ݱൣ 6ݱݲ ൢ 9ݲቘ
Note: Resulting trinomial does not factor.
D. Difference of two cubes:
ൣ 64 ൩ݱ ൣ 4ቘݱൢ 4ݱ ൢ 16ቘNote: 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:
ൢ 6ݱ ൢ 9 ൩ݱ ൢ 3ቘݱ ൢ 3ቘ൩ݱ ൢ 3ቘ
4ݱ ൣ 20ݱݲ ൢ 25ݲ൩2ݱ ൣ 5ݲቘ ൢ 7ݱ ൢ 12 ൩ݱ ൢ 3ቘݱ ൢ 4ቘ ൣ 7ݱ ൢ 12 ൩ݱ ൣ 3ቘݱ ൣ 4ቘ ൢ 3ݱ ൣ 18 ൩ݱ ൢ 6ቘݱ ൣ 3ቘ ൣ 3ݱ ൣ 18 ൩ݱ ൣ 6ቘݱ ൢ 3ቘ4) If factoring a polynomial with four terms, possible choices are below.
A. Group first two terms together and last two terms together.ൣ 3ݱൢ 2ݱ ൣ 6 ൩ݱൣ 3ݱቘൢ2ݱ ൣ 6ቘ൩ ݱݱ ൣ 3ቘൢ 2ݱ ൣ 3ቘ൩ݱ ൣ 3ቘݱൢ 2ቘ
B. Group first three terms together.
ݱൢ 6ݱ ൢ 9 ൣ ݲ൩ݱൢ 6ݱ ൢ 9ቘൣ ݲ൩ݱ ൢ 3ቘൣ ݲ൩ቛݱ ൢ 3ቘൢ ݲቜቛݱ ൢ 3ቘൣ ݲቜ൩ݱ ൢ 3 ൢ ݲቘݱ ൢ 3 ൣ ݲቘ
C. Group last three terms together.
ݲൣ ݱൢ 6ݱ ൣ 9 ൩ ݲൣݱൣ 6ݱ ൢ 9ቘ൩ ݲൣݱ ൣ 3ቘ൩ቛݲ ൢݱ ൣ 3ቘቜቛݲ ൣݱ ൣ 3ቘቜ൩ݲ ൢ ݱ ൣ 3ቘݲ ൣ ݱ ൢ 3ቘ
BE SURE YOUR ANSWERS WILL NOT FACTOR FURTHER!
All answers may be checked by multiplication.
PRACTICE PROBLEMS:
1. ݲൢ 9ݲ
2.5ݱݲൢ 15ݱݲ
3.12ݭୖൣ 20ݭ୕ൢ 8ݭൣ 16
4.ݩൣ 36
5.25 ൣ ݱ
6. 7. 8.9 ൣݱ ൣ ݲቘ
9.ݲൢ 8
10.64ݲ୕ൢ ݲ
11.ݱൣ 27
12.5ݱൣ 40ݲ
13.2ݲ୕ൣ 128ݲ
14.ݭୗൣ 64
15.ݱൣ 10ݱ ൢ 25
16. 17.16ݲൢ 56ݲ ൢ 49
18. ൣ20ݱݲ ൢ 4ݲൢ 25ݱ 19.ݱൢ 9ݱ ൢ 20
20.2ݲൣ 16ݲ ൢ 32 21.
3ݱ ൢ ݱൣ 10
22.ݲൢ 5ݲ ൣ 84
23.8ݱൣ 16 ൣ 28ݱ
24.12ݱൣ 31ݱൢ 20ݱ
25.26.
8 ൣ 6ݱ ൣ 9ݱ
27.6ݱୗൢ ݱൣ 2
28.2ݱൣ 14ݱ୕ൢ 20
29.ݲൣ ݲൢ 2ݲ ൣ 2
30.31.
ݱൢ 8ݱൣ ݱ ൣ 8
32.ݩݪ ൣ 25ݪ ൢ 3ݩൣ 75 33.
16 ൣ ݱൢ 2ݱݲ ൣ ݲ
34.2ݱݲ ൣ ݱݲ ൣ 6 ൢ 3ݱ
35.6ݱൢ 23ݱ ൢ 20
36.9ݱൢ 15ݱ ൢ 4
37.8ݦൣ 6ݦ ൣ 9
38.25 ൣ 10ݱ ൢ ݱ
39.16 ൣ ݰ୕
40.ANSWERS:
1.ݲݲ ൢ 9ቘ 2. 5ݱݲݲ ൢ 3ݱቘ 3. 43ݭୖൣ 5ݭ୕ൢ 2ݭൣ 4ቘ 4. ݩ ൢ 6ቘݩ ൣ 6ቘ
5. 8.3 ൢ ݱ ൣ ݲቘ3 ൣ ݱ ൢ ݲቘ 9. ݲ ൢ 2ቘݲൣ 2ݲ ൢ 4ቘ 10. ݲ4ݲ ൢ 1ቘ16ݲൣ 4ݲ ൢ 1ቘ
11.ݱ ൣ 3ቘݱൢ 3ݱ ൢ 9ቘ 12. 5ݱ ൣ 2ݲቘݱൢ 2ݱݲ ൢ 4ݲቘ 13. 2ݲݲ ൣ 4ቘݲൢ 4ݲ ൢ 16ቘ
14. 18.5ݱ ൣ 2ݲቘ 19. ݱ ൢ 5ቘݱ ൢ 4ቘ 20. 2ݲ ൣ 4ቘ 21. ݱ ൢ 5ቘݱ ൣ 2ቘ 22. ݲ ൢ 12ቘݲ ൣ 7ቘ
23.27.
3ݱൢ 2ቘ2ݱൣ 1ቘ 28. 2ݱ୕ൣ 5ቘݱ୕ൣ 2ቘ 29. ݲ ൣ 1ቘݲൢ 2ቘ 30. ݱݱൢ 1ቘݱ ൣ 1ቘ
31.ݱ ൢ 8ቘݱ ൢ 1ቘݱ ൣ 1ቘ 32. ݪ ൢ 3ቘݩ ൢ 5ቘݩ ൣ 5ቘ 33. 4 ൢ ݱ ൣ ݲቘ4 ൣ ݱ ൢ ݲቘ
34.2 ൣ ݱቘݱݲ ൣ 3ቘ 35. 3ݱ ൢ 4ቘ2ݱ ൢ 5ቘ 36. 3ݱ ൢ 1ቘ3ݱ ൢ 4ቘ 37. 4ݦ ൢ 3ቘ2ݦ ൣ 3ቘ
38.MORE PRACTICE PROBLEMS:
41. ݱൣ 6ݱ ൣ 16
42.ݱൣ 10ݱݲ ൢ 24ݲ
43.ݱൢ 3ݱ ൢ 2
44.ݱൣ 3ݱ ൢ 2
45.ݱൣ ݱ ൣ 30
46.ݱൢ 7ݱ ൣ 8
47.ݱൢ ݱ ൣ 2
48.ݱൣ 5ݱݲ ൢ 6ݲ
49.ݱൢ 10ݱ ൢ 16
50.ݱൢ ݱ ൣ 72
51.ݱൣ 8ݱ ൣ 9
52.ݱൢ 2ݱ ൣ 48
53.ݱൣ 13ݱݲ ൢ 42ݲ
54.ݱൢ 8ݱ ൢ 12
55.4ݱൣ 8ݱൣ 12ݱ
56.2ݱൣ 2ݱൣ 4ݱ
57.2ݱൣ 4ݱൣ 6ݱ
58.3ݱൣ 6ݱൣ 9ݱ
59.5ݱݲ ൣ 35ݱݲ ൢ 50ݱݲ
60.3ݱݲ ൢ 18ݱݲ ൣ 21ݱݲ 61.
4ݱൢ 1 ൣ 4ݱ
62.15ݱൢ 12 ൢ 29ݱ
63.8ݫൣ 2ݫ ൣ 3
64.65.
25ݱൢ 8 ൢ 30ݱ
66.12ݱൢ 3 ൢ 13ݱ
67.9ݱൣ 27ݱݲ ൢ 20ݲ
68.25ݮൣ 15ݮ ൣ 18
69.12ݟൣ 4ݟ ൣ 5
70.5ݳൢ 3ݳ ൢ 4
71.4ݱൢ 15 ൢ 16ݱ
72.20ݱൢ 6 ൢ 23ݱ
73.6ݱൣ 19ݱݲ ൢ 10ݲ
74.35ݩൢ 13ݩ ൣ 4
75.50ݱൢ 10ݱ ൣ 12
76.ൣ30ݱൣ 25ݱ ൢ 30 77.
ൣ18ݱൢ 18ݱ ൢ 20 78.
3ݱൣ 22ݱൢ 7ݱ
79.15ݱൣ 18ݱ ൣ 24
80.4ݱൣ 25ݱൢ 6ݱ
ANSWERS:
41.ݱ ൣ 8ቘݱ ൢ 2ቘ 42. ݱ ൣ 6ݲቘݱ ൣ 4ݲቘ 43. ݱ ൢ 2ቘݱ ൢ 1ቘ 44. ݱ ൣ 2ቘݱ ൣ 1ቘ
45.ݱ ൣ 6ቘݱ ൢ 5ቘ 46. ݱ ൢ 8ቘݱ ൣ 1ቘ 47. ݱ ൢ 2ቘݱ ൣ 1ቘ 48. ݱ ൣ 3ݲቘݱ ൣ 2ݲቘ
49.ݱ ൢ 8ቘݱ ൢ 2ቘ 50. ݱ ൢ 9ቘݱ ൣ 8ቘ 51. ݱ ൣ 9ቘݱ ൢ 1ቘ 52. ݱ ൢ 8ቘݱ ൣ 6ቘ
53.ݱ ൣ 7ݲቘݱ ൣ 6ݲቘ 54. ݱ ൢ 6ቘݱ ൢ 2ቘ 55. 4ݱݱ ൣ 3ቘݱ ൢ 1ቘ 56. 2ݱݱ ൣ 2ቘݱ ൢ 1ቘ
57.2ݱݱ ൣ 3ቘݱ ൢ 1ቘ 58. 3ݱݱ ൣ 3ቘݱ ൢ 1ቘ 59. 5ݱݲݱ ൣ 5ቘݱ ൣ 2ቘ 60. 3ݱݲݱ ൢ 7ቘݱ ൣ 1ቘ 61.
65.5ݱ ൢ 4ቘ5ݱ ൢ 2ቘ 66. 3ݱ ൢ 1ቘ4ݱ ൢ 3ቘ 67. 3ݱ ൣ 5ݲቘ3ݱ ൣ 4ݲቘ 68. 5ݮ ൢ 3ቘ5ݮ ൣ 6ቘ 69.
72.5ݱ ൢ 2ቘ4ݱ ൢ 3ቘ 73. 2ݱ ൣ 5ݲቘ3ݱ ൣ 2ݲቘ 74. 7ݩ ൢ 4ቘ5ݩ ൣ 1ቘ
75.25ݱ ൢ 3ቘ5ݱ ൣ 2ቘ 76. ൣ52ݱ ൢ 3ቘ3ݱ ൣ 2ቘ 77. ൣ23ݱ ൣ 5ቘ3ݱ ൢ 2ቘ
78.ݱ3ݱ ൣ 1ቘݱ ൣ 7ቘ 79. 35ݱ ൢ 4ቘݱ ൣ 2ቘ 80. ݱ4ݱ ൣ 1ቘݱ ൣ 6ቘ
MORE PRACTICE PROBLEMS:
81. 125ݱൣ 1
82.ݰൣ 64
83.ݲൣ 12ݲ ൢ 36
84.ݱൣ 8ݱ ൣ 48
85.86.
87.
ݱ ൣ 3ቘݱ ൢ 7ቘൢݱ ൣ 3ቘݱ ൣ 4ቘ
88.6ݱൢ 12ݱ ൢ 6
89.ݲൣ 11ݲ ൢ 18
90.40 ൢ 3ݛ ൣ ݛ
91.3ݱୖൣ 12ݱ
92.250ݱൢ 2
93.7ݱݲ୕ൣ 7ݱݳ୕
94.2ݲ୕ൢ 5ݲൣ 12ݲ
95.24ݱൣ 7ݱ ൣ 5
96.ݲൢ 14ݲ ൣ 32
97.0.04ݰൢ 0.28ݰ ൢ 0.49
98.4ݱൢ 40ݱൢ 64ݱ
99.64ݲൢ 27
100.ൣ ݱ 101.
5ݱൣ 2ݱ ൢ 3
102.ݱൣ 343
103.40ݲൢ 28ݲ ൣ 48
104.105.
8ݜୗൣ 125ݝୗ
106.81 ൣ 18ݳ ൢ ݳ
107.ݱ୕ൢ 10ݱൢ 25ݱ
108.109.
ݲൢ 5ݲ ൣ 36
110.ݱൣ 11ݱ ൣ 42
111.112.
113.
81 ൢ 18ݲ ൢ ݲ
114.ݛൣ 5ݛ ൣ 14
115.ݪ୕ൣ 10ݪൢ 21ݪ
116.9ݱݲൣ 25ݲ୕
117.105 ൢ 8ݱ ൣ ݱ
118.ݱൣ 3ݱ ൣ 2
119.6ݲൢ 48
120.ANSWERS:
81.5ݱ ൣ 1ቘ25ݱൢ 5ݱ ൢ 1ቘ 82. ݰ ൢ 8ቘݰ ൣ 8ቘ 83. ݲ ൣ 6ቘ 84. ݱ ൣ 12ቘݱ ൢ 4ቘ
88. 6ݱ ൢ 1ቘ
89. ݲ ൣ 9ቘݲ ൣ 2ቘ 90. 8 ൣ ݛቘ5 ൢ ݛቘ 91. 3ݱݱൣ 4ቘ
92. 25ݱ ൢ 1ቘ25ݱ
ൣ 5ݱ ൢ 1ቘ 93. 7ݱݲൢ ݳቘݲ ൢ ݳቘݲ ൣ ݳቘ 94. ݲ2ݲ ൣ 3ቘݲ ൢ 4ቘ
95.8ݱ ൣ 5ቘ3ݱ ൢ 1ቘ 96. ݲ ൣ 2ቘݲ ൢ 16ቘ 97. 0.2ݰ ൢ 0.7ቘ
98. 4ݱݱ ൢ 2ቘݱ ൢ 8ቘ 99.4ݲ ൢ 3ቘ16ݲ ൣ 12ݲ ൢ 9ቘ 100. ቝ
ݱ ൣ 7ቘݱ
105.2ݜ
ൣ 5ݝቘ4ݜ୕ൢ 10ݜݝൢ 25ݝ୕ቘ 106. 9 ൣ ݳቘ 107. ݱݱ ൢ 5ቘ
108.112.
ቘ 113. 9 ൢ ݲቘ 114. ݛ ൣ 7ቘݛ ൢ 2ቘ 115. ݪݪ ൣ 3ቘݪ ൣ 7ቘ 116.
119. 6ݲ ൢ 2ቘݲ
121.3ݲൣ 34ݲ ൣ 24
122.123.
ݲൣ 121
124.125.
9ݱൣ 24ݱൢ 16ݱ
126.127.
10ݰൢ 29ݰ ൣ 21
128.16ݱൢ 54ݱ ൣ 7
129.27ݱൣ 30ݱ ൣ 8
130.ݱୗൣ 1 131.
ݱൣ 0.6ݱ ൢ 0.09
132.4ݱൣ 13ݱ ൣ 35
133.quotesdbs_dbs17.pdfusesText_23