Factoring Worksheet
Math 0097 Elementary and Intermediate Algebra
Notes
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To factor a polynomial completely, look for:
- greatest common factor, GCF
- a perfect square trinomial
- a different of two squares
- a pair of binomial factors
- a grouping of terms
- Factor it out if there is a common factor.
- If it is binomial (two terms), check if it is the difference of two squares.
- 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.
- If the polynomial has more than three terms, try to factor by using the grouping of terms.
- Check that each factor is prime.
- Check your answer by multiplying all the factors.
The following exercises include various kinds of factoring methods:
- 32A9 + 16A5 + 24A3
- 2A2 – 10A – 48
- 14M3 + 7M2 – 70M
- 9X2 + 64
- 6X2 + 31X + 5
- A2 – 3AB – 4B2
- 49X2 – 16Y2
- 100B2C2 + 30BC3 – 50B2C
- 16X + 20
- 2A2 + 5A – 3
- 10Y2 – 7YX –6X2
- Y4 – 16
- A2 + 2A –15
- 6Y2 – 5Y – 4
- 32X3 +56X2 –16X
- 15Y + 5
- X2 – 12X + 36
- 9A2 – 64
- Y2 – 4YM –12M2
- 16X2 – 8X + 1
- 6Y2 – 6Y – 12
- 72Y3X2 + 12Y2 – 24Y4X2
- E2 – 17E + 66
- A2 + 17A + 72
- M2 + 9
- 108A2 – 36A + 3
- X2 – 3XA –10A2
- 45A3B5 – 60A4B2 + 75A5B4
- 4M2 – 12M + 9
- A2 – 3AB –28B2
- 16C2 + 24CA + 9A2
- 3M2 + 4M – 4
- 3M3 – 12M2 – 15M
- A4 – 625
- 16M2 – 48M + 36
- 8M2 –10M – 3
- 36Y6 – 42Y5 –120Y4
- 8E2 + 23E – 3
- 5X3 – 45X2 + 70X
- 8M2 – 2MN – 3N2
- 54A2 – 24X2
- 4M2 – 20MX + 25X2
- 6A2 +10A – 4
- 15N2 + 11NE –14E2
- A2 – 81
- 10X2 – 7X – 6
- 125A4 – 400A3B + 195A2B2
- 9Y2 + 12Y – 5
- A2 – 4A + 4
- 27E10 – 45E9 – 252E8
- 24M4E + 60M3E2 + 125M2E3
- 10A2 +25A – 60
- 12E2 + EQ – 6Q2
- M2 – 64
- 64E2 – 100A2
- 2A2 + 7AB – 15B2
- 100A2 – 81Y2
- 8A2 + 23AB – 3B2
- A2 + 8A + 16
- 4Y2 – 25
- X2 = 64
- 2X+4
- 6 X2 – 3x – 4x + 2
- X2 + 3X + 2
- A3 – B3
- A3 – 8
- A3 + B3
- A3 +8
- X2 + 2X + 1
- X2 – 2X + 1
Answers:
- 32A9 + 16A5 + 24A3 = 8A3(4A6 + 2A2 + 3)
- 2A2 – 10A – 48 = 2(A2 – 5A – 24)= 2 (A – 8)(A + 3)
- 14M3 + 7M2 – 70M = 7M(2M + 5)(M – 2)
- 9X2 + 64 Prime
- 6X2 + 31X + 5 = (X + 5 )6X + 1)
- A2 – 3AB – 4B2 = (A – 4B)(A + B)
- 49X2 – 16Y2 = (7X + 4Y)(7X – 4Y)
- 100B2C2 + 30BC3 – 50B2C = 10BC(10BC + 3C2 – 5B)
- 16X + 20 = 4(4X + 5)
- 2A2 + 5A – 3 = (A + 3)(2A –1)
- 10Y2 – 7YX –6X2 = (5Y – 6X)(2X + Y)
- Y4 – 16 = (Y2 + 4)(Y + 2)(Y – 2)
- A2 + 2A –15 = (A + 5)(A – 3)
- 6Y2 – 5Y – 4 = (3X – 4)(2X + 1)
- 32X3 + 56X2 –16X = 8X(X + 2)(4X – 1)
- 15Y + 5 = 5(3Y + 1)
- X2 – 12X + 36 = (X – 6)(X – 6) = (X – 6)2
- 9A2 – 64 = (3A + 8)(3A – 8)
- Y2 – 4YM –12M2 = (Y – 6M)(Y + 2M)
- 16X2 – 8X + 1 = (4X – 1)(4X –1) = (4X –1)2
- 6Y2 – 6Y – 12 = 6(Y – 2)(Y + 1)
- 72Y3X2 + 12Y2 – 24Y4X2 = 12Y2(6YX2 + 1 – 2Y2X2)
- E2 – 17E + 66 = (E – 11)(E – 6)
- A2 + 17A + 72 = (A + 8)(A + 9)
- M2 + 9 Prime
- 108A2 – 36A + 3 = 3(36A2 – 12A + 1) =3(6A –1)(6A – 1)
- X2 – 3XA –10A2 = (X – 5A)(X + 2A)
- 45A3B5 – 60A4B2 + 75A5B4 = 15A3B2 (3B3 – 4A + 5A2B2)
- 4M2 – 12M + 9 = (2M –3)2
- A2 – 3AB –28B2 = (A – 7B)(A + 4B)
- 16C2 + 24CA + 9A2 = (4C + 3A)2
- 3M2 + 4M – 4 = (M + 2)(3M – 2)
- 3M3 – 12M2 – 15M = 3M(M – 5)(M + 1)
- A4 –625 =(A2 + 25)(A + 5)(A – 5)
- 16M2 – 48M + 36 = (4M – 6)2
- 8M2 –10M – 3 = (2M – 3)(4M + 1)
- 36Y6 – 42Y5 –120Y4 = 6Y4 (6Y2 – 7Y – 20) = 6Y4 (2Y – 5)(3Y + 4)
- 8E2 + 23E – 3 = (E + 3)(8E – 1)
- 5X3 – 45X2 + 70X = 5X(X2 – 9X + 14) = 5X(X – 7) (X – 2)
- 8M2 – 2MN – 3N2 = (4M – 3N)(2M + N)
- 54A2 – 24X2 = 6(9A2 – 4X2) = 6(3A +2X) (3A – 2X)
- 4M2 – 20 MX + 25X2 = (M – 5X)2
- 6A2 + 10A – 4 = 2(3A2 + 5A – 2) =2(3A – 1) (A + 2)
- 15N2 + 11NE –14E2 = (5N + 7)(3N – 2)
- A2 – 81 = (A + 9)(A – 9)
- 10X2 – 7X – 6 = (2X + 1)(5X – 6)
- 125A4 – 400A3B + 195A2B2 = 5A2 (25A2 – 80AB + 39B2) =5A2 (5A – 13B) (5A – 3B)
- 9Y2 + 12Y – 5 = (3Y + 5)(3Y –1)
- A2 – 4A + 4 = (A – 2)(A – 2) = (A – 2)2
- 27E10 – 45E9 – 252E8 = 9E8(3E2 – 5E – 28) = 9E8 (3E + 7) (E – 1)
- 24M4E + 60M3E2 + 125M2E3 = M2E (24M2 + 60ME + 125E2)
- 10A2 + 25A – 60 = 5(2A2 + 5A – 12) = 5(A + 4) (2A – 3)
- 12E2 + EQ – 6Q2 = (4E + 3)(3E –2)
- M2 – 64 = (M + 8)(M – 8)
- 64E2 – 100A2 = 4(16E2 – 25A2) = 4(4E + 5A) (4E – 5A)
- 2A2 + 7AB – 15B2 = (2A – 3B)(A + 5B)
- 100A2 – 81Y2 = (10A + 9Y)(10A – 9Y)
- 8A2 + 23AB – 3B2 = (8A – B)(A + 3B)
- A2 + 8A + 16 = (A + 4)(A + 4) = (A + 4 )2
- 4Y2 – 25 = (2Y + 5)(2Y – 5)
- X2 = 64 = X2 – 64 = (X-8) (X+8)
- 2X+4 = 2 (X+2)
- 6 X2 – 3x – 4x + 2 = (6 X2 – 3x) + (– 4x + 2) = 3x (2x – 1) + -2(2x – 1) = (2x – 1) (3x – 2)
- X2 + 3X + 2 = (X+1)(X+2)
- A3 – B3 = (A – B) (A2 + AB + B2 )
- A3 – 8 = A3 – 23 = (A – 2) (A2 + 2A + 4 )
- A3 + B3 = (A + B) (A2 – AB + B2 )
- A3 + 8 = A3 + 23 = (A + 2) (A2 – 2A + 4 )
- X2 + 2X + 1 = (X+1)(X+1) = (X+1)2
- X2 – 2X + 1 = (X-1)(X-1) = (X-1)2