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# Factoring Worksheet

Math 0097  Elementary and Intermediate Algebra

Notes

1. To factor a polynomial completely, look for:

1. greatest common factor, GCF
2. a perfect square trinomial
3. a different of two squares
4. a pair of binomial factors
5. a grouping of terms
2. Factor it out if there is a common factor.
3. If it is binomial (two terms), check if it is the difference of two squares.
4.  If it is trinomial (three terms), check if it is a perfect square trinomial.  If is not a perfect square, check if the coefficient of squared term is 1 then use the method in section 5.2.  If the coefficient of the squared terms is not equal to 1, use general factoring methods.
5. If the polynomial has more than three terms, try to factor by using the grouping of terms.
6. Check that each factor is prime.
7. Check your answer by multiplying all the factors.

The following exercises include various kinds of factoring methods:

1.  32A9 + 16A5 + 24A3
2.  2A2 – 10A – 48
3.  14M3 + 7M2 – 70M
4.  9X2 + 64
5.  6X2 + 31X + 5
6.  A2 – 3AB – 4B2
7.  49X2 – 16Y2
8.  100B2C2 + 30BC3 – 50B2C
9.    16X + 20
10.  2A2 + 5A – 3
11.  10Y2 – 7YX –6X2
12.  Y4 – 16
13.  A2 + 2A –15
14.  6Y2 – 5Y – 4
15. 32X3 +56X2 –16X
16.  15Y + 5
17. X2 – 12X + 36
18. 9A2 – 64
19. Y2 – 4YM –12M2
20. 16X2 – 8X + 1
21.  6Y2 – 6Y – 12
22.  72Y3X2 + 12Y2 – 24Y4X2
23.  E2 – 17E + 66
24. A2 + 17A + 72
25. M2 + 9
26. 108A2 – 36A + 3
27.  X2 – 3XA –10A2
28. 45A3B5 – 60A4B2 + 75A5B4
29. 4M2 – 12M + 9
30. A2 – 3AB –28B2
31. 16C2 + 24CA + 9A2
32. 3M2 + 4M – 4
33. 3M3 – 12M2 – 15M
34.  A4 – 625
35. 16M2 – 48M + 36
36. 8M2 –10M – 3
37. 36Y6 – 42Y5 –120Y4
38. 8E2 + 23E – 3
39. 5X3 – 45X2 + 70X
40. 8M2 – 2MN – 3N2
41. 54A2 – 24X2
42. 4M2 – 20MX + 25X2
43. 6A2 +10A – 4
44.  15N2 + 11NE –14E2
45.  A2 – 81
46. 10X2 – 7X – 6
47.  125A4 – 400A3B + 195A2B2
48. 9Y2 + 12Y – 5
49. A2 – 4A + 4
50. 27E10 – 45E9 – 252E8
51. 24M4E + 60M3E2 + 125M2E3
52. 10A2 +25A – 60
53. 12E2 + EQ – 6Q2
54. M2 – 64
55. 64E2 – 100A2
56. 2A2 + 7AB – 15B2
57. 100A2 – 81Y2
58. 8A2 + 23AB – 3B2
59. A2 + 8A + 16
60. 4Y2 – 25
61.  X2 = 64
62. 2X+4
63. 6 X2 – 3x – 4x + 2
64. X2  + 3X + 2
65. A3 – B3
66. A3 – 8
67. A3 + B3
68. A3 +8
69. X2  + 2X + 1
70. X2  – 2X + 1

Answers:

1.  32A9 + 16A5 + 24A3 = 8A3(4A6 + 2A2 + 3)
2.  2A2 – 10A – 48 = 2(A2 – 5A – 24)= 2 (A – 8)(A + 3)
3.  14M3 + 7M2 – 70M = 7M(2M + 5)(M – 2)
4.  9X2 + 64  Prime
5.  6X2 + 31X + 5 = (X + 5 )6X + 1)
6. A2 – 3AB – 4B2 = (A – 4B)(A + B)
7. 49X2 – 16Y2 = (7X + 4Y)(7X – 4Y)
8. 100B2C2 + 30BC3 – 50B2C = 10BC(10BC + 3C2 – 5B)
9. 16X + 20 = 4(4X + 5)
10.  2A2 + 5A – 3 = (A + 3)(2A –1)
11.  10Y2 – 7YX –6X2 = (5Y – 6X)(2X + Y)
12.  Y4 – 16 = (Y2 + 4)(Y + 2)(Y – 2)
13.  A2 + 2A –15 = (A + 5)(A – 3)
14.  6Y2 – 5Y – 4 = (3X – 4)(2X + 1)
15. 32X3 + 56X2 –16X = 8X(X + 2)(4X – 1)
16.  15Y + 5 = 5(3Y + 1)
17. X2 – 12X + 36 = (X – 6)(X – 6) = (X – 6)2
18. 9A2 – 64 = (3A + 8)(3A – 8)
19. Y2 – 4YM –12M2 = (Y – 6M)(Y + 2M)
20. 16X2 – 8X + 1 = (4X – 1)(4X –1) = (4X –1)2
21.  6Y2 – 6Y – 12 = 6(Y – 2)(Y + 1)
22.  72Y3X2 + 12Y2 – 24Y4X2 = 12Y2(6YX2 + 1 – 2Y2X2)
23.  E2 – 17E + 66 = (E – 11)(E – 6)
24. A2 + 17A + 72 = (A + 8)(A + 9)
25. M2 + 9  Prime
26. 108A2 – 36A + 3 = 3(36A2 – 12A + 1) =3(6A –1)(6A – 1)
27.  X2 – 3XA –10A2 = (X – 5A)(X + 2A)
28. 45A3B5 – 60A4B2 + 75A5B4 = 15A3B2 (3B3 – 4A + 5A2B2)
29. 4M2 – 12M + 9 = (2M –3)2
30.  A2 – 3AB –28B2 = (A – 7B)(A + 4B)
31. 16C2 + 24CA + 9A2 = (4C + 3A)2
32.  3M2 + 4M – 4 = (M + 2)(3M – 2)
33.  3M3 – 12M2 – 15M = 3M(M – 5)(M + 1)
34.  A4 –625 =(A2 + 25)(A + 5)(A – 5)
35. 16M2 – 48M + 36 = (4M – 6)2
36. 8M2 –10M – 3 = (2M – 3)(4M + 1)
37. 36Y6 – 42Y5 –120Y4 = 6Y4 (6Y2 – 7Y – 20) = 6Y4 (2Y – 5)(3Y + 4)
38. 8E2 + 23E – 3 = (E + 3)(8E – 1)
39. 5X3 – 45X2 + 70X = 5X(X2 – 9X + 14) = 5X(X – 7) (X – 2)
40. 8M2 – 2MN – 3N2 = (4M – 3N)(2M + N)
41. 54A2 – 24X2 = 6(9A2 – 4X2) = 6(3A +2X) (3A – 2X)
42. 4M2 – 20 MX + 25X2 = (M – 5X)2
43. 6A2 + 10A – 4 = 2(3A2 + 5A – 2) =2(3A – 1) (A + 2)
44.  15N2 + 11NE –14E2 = (5N + 7)(3N – 2)
45.  A2 – 81 = (A + 9)(A – 9)
46. 10X2 – 7X – 6  = (2X + 1)(5X – 6)
47.  125A4 – 400A3B + 195A2B2 = 5A2 (25A2 – 80AB + 39B2) =5A2 (5A – 13B) (5A – 3B)
48. 9Y2 + 12Y – 5 = (3Y + 5)(3Y –1)
49. A2 – 4A + 4 = (A – 2)(A – 2) = (A – 2)2
50. 27E10 – 45E9 – 252E8 = 9E8(3E2 – 5E – 28) = 9E8 (3E + 7) (E – 1)
51. 24M4E + 60M3E2 + 125M2E3 = M2E (24M2 + 60ME + 125E2)
52. 10A2 + 25A – 60 = 5(2A2 + 5A – 12) = 5(A + 4) (2A – 3)
53. 12E2 + EQ – 6Q2 = (4E + 3)(3E –2)
54. M2 – 64 = (M + 8)(M – 8)
55. 64E2 – 100A2 = 4(16E2 – 25A2) = 4(4E + 5A) (4E – 5A)
56. 2A2 + 7AB – 15B2 = (2A – 3B)(A + 5B)
57. 100A2 – 81Y2 = (10A + 9Y)(10A – 9Y)
58. 8A2 + 23AB – 3B2 = (8A – B)(A + 3B)
59. A2 + 8A + 16 = (A + 4)(A + 4) = (A + 4 )2
60. 4Y2 – 25 = (2Y + 5)(2Y – 5)
61. X2 = 64 = X2 – 64 = (X-8) (X+8)
62. 2X+4 = 2 (X+2)
63. 6 X2 – 3x – 4x + 2 = (6 X2 – 3x) + (– 4x + 2) = 3x (2x – 1) + -2(2x – 1) = (2x – 1) (3x – 2)
64. X2  + 3X + 2 = (X+1)(X+2)
65. A3 – B3  = (A – B) (A2 + AB + B2 )
66. A3 – 8 = A3 – 23 = (A – 2) (A2 + 2A + 4 )
67. A3 + B3 = (A + B) (A2 – AB + B2 )
68. A3 + 8  = A3 + 23 = (A + 2) (A2 – 2A + 4 )
69. X2  + 2X + 1 = (X+1)(X+1) = (X+1)2
70. X2  – 2X + 1 = (X-1)(X-1) = (X-1)2