GHC Home > Dilshad Akrayee > Math 987 > Extra Support > Factoring Worksheet

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