Factoring Polynomials
Factoring Binomials:
*Always look for a GCF (greatest common factor) first.
Difference of Squares: A2 – B2 = (A + B)(A – B) identify the square roots
EX: 50x2 – 8 = 2 (25x2 – 4) = 2(5x + 2)(5x – 2) GCF & Difference of Squares
Sum of Squares: A2 + B2 = PRIME … will never be factorable unless there is a GCF
EX: 25x2 + 15 = 5(5x2 + 3) It cannot be factored any further.
Difference of CUBES: A3 – B3 = (A – B)(A2 + AB + B2) identify the cube roots
EX: 27x3 – 1 = (3x)3 – (1)3 = (3x – 1)[(3x)2 + (3x)(1) + (1)2] = (3x – 1)(9x2 + 3x + 1)
EX: 8x3 – 125 = (2x)3 – (5)3 = (2x – 5)[(2x)2 + (2x)(5) + (5)2] = (2x – 5)(4x2 + 10x + 25)
*Note: The quadratic part of the pattern should not be factorable.
Sum of CUBES: A3 + B3 = (A + B)(A2 – AB + B2) identify the cube roots
EX: 81x3 + 3 = 3(27x3 + 1) = 3[(3x)3 + (1)3] ... GCF
= 3(3x + 1)[(3x)2 – (3x)(1) + (1)2] = 3(3x + 1)(9x2 – 3x + 1)
EX: 64x3 + 27 = (4x)3 + (3)3 = (4x + 3)[(4x)2 – (4x)(3) + (3)2] = (4x + 3)(16x2 – 12x + 9)
*Note: The quadratic part of the pattern should not be factorable.
Factoring by Grouping: (4 terms or 6 terms)
AC + AD + BC + BD … look for the GCF between each PAIR of terms
= A(C + D) + B(C + D) … Factor out the GCF of each pair. Do the parentheses match?
= (A + B)(C + D) …The GCF’s make up one factor and the parentheses is the other!
EX: x3 + 4x2 + 3x + 12
= x2(x + 4) + 3(x + 4) GCF of the pairs
= (x2 + 3)(x + 4) GCF’s & Parentheses
EX: 2x3 + 6x2 – x – 3
= 2x2(x + 3) – 1(x + 3) (Divide out the -1)
= (2x2 – 1)(x + 3)
-------------------------------------------------------------------------------------------------------D6. Factor each Polynomial Completely Name______________________
1. 36x2 – 81
6. x3 + 27
2. 8x2 – 200
7. 125x3 + 8
3. 50x2 + 36
8. x3 + 2x2 + x + 2
4. 27x3 – 125
9. 3x3 – 12x2 – 2x + 8
5. 250x3 – 2
10. x3 + 2x2 – 4x – 8