Guidelines for Factoring Completely
1. Descending order. If the polynomial contains two variables, put in descending order according to the
variable that comes first in the alphabet.
2. Common factor. Make sure you have factored out the largest common factor.
3. Factor. Look at the number and type of terms to figure out which method to use.
TWO TERMS
Difference of Squares
A – B2 = (A – B)(A + B)
Difference of Cubes
A – B3 = (A – B)(A2 + AB + B2)
Sum of Cubes
A + B = (A + B)(A2 – AB + B2)
x2 – 25 = (x – 5)(x + 5)
4x2 – 9 = (2x – 3)(2x + 3)
x3 – 27 = (x – 3)(x2 + 3x + 9)
8x – 125 = (2x – 5)(4x2 + 10x +25)
x3 + 64 = (x + 4)(x2 – 4x + 16)
216x3 + 1 = (6x + 1)(36x2 – 6x + 1)
2
3
3
3
3
THREE TERMS
x2 + bx + c
Trial and Error
ax2 + bx + c
Trial and Error or Box Method
Perfect Square Trinomial
A2 + 2AB + B2 = (A + B)2
A2 – 2AB + B2 = (A – B)2
x2 + 10x + 16 = (x + 2)(x + 8)
x2 – 10x + 21 = (x – 3)(x – 7)
6x2 + 19x + 15 = (2x + 3)(3x + 5)
3x2 – 25x + 42 = (x – 6)(3x – 7)
x2 + 16x + 64 = (x + 8) 2
16x2 – 56x + 49 = (4x – 7) 2
FOUR TERMS
Grouping or Box Method
6x3 + 8x2 + 15x + 20 = (2x2 + 5)(3x + 4)
4x3 + 12x2 – 15x – 15 = (4x2 – 5)(x + 3)
4. Factor Again. See if any factor can be factored again using one of the above steps. Keep factoring until
you can not factor anymore
x4 – 16 = (x2 + 4) (x2 – 4) = (x2 + 4)(x – 2)(x + 2)
(2x – 10)(x + 4) = 2(x – 5)(x + 4)
5. Check. Multiply or evaluate to check your answer.
Definition: A prime polynomial is polynomial that can not be factored.
PRIME
4x2 + 9
x2 – 5
x2 + 81
x2 + 8x + 5