Factoring Rules & Patterns — Complete Algebra Reference A complete reference of factoring patterns and techniques. Includes the standard rules every algebra student needs: greatest common factor, difference of squares, factoring trinomials, factoring by grouping, sum and difference of cubes, and a step-by-step flowchart for choosing the right method.
Greatest Common Factor (GCF) Factor Out GCF
a b + a c = a ( b + c ) ab + ac = a(b + c) ab + a c = a ( b + c ) GCF with Negative
− a x − a y = − a ( x + y ) -ax - ay = -a(x + y) − a x − a y = − a ( x + y ) Multivariable GCF
6 x 3 y 2 + 9 x 2 y = 3 x 2 y ( 2 x y + 3 ) 6x^3 y^2 + 9 x^2 y = 3 x^2 y (2 x y + 3) 6 x 3 y 2 + 9 x 2 y = 3 x 2 y ( 2 x y + 3 ) Step 1 of Every Factoring
Always factor out the GCF first \text{Always factor out the GCF first} Always factor out the GCF first Difference of Squares Pattern
a 2 − b 2 = ( a + b ) ( a − b ) a^2 - b^2 = (a + b)(a - b) a 2 − b 2 = ( a + b ) ( a − b ) With Variables
x 2 − 9 = ( x + 3 ) ( x − 3 ) x^2 - 9 = (x + 3)(x - 3) x 2 − 9 = ( x + 3 ) ( x − 3 ) Two Variables
x 2 − y 2 = ( x + y ) ( x − y ) x^2 - y^2 = (x + y)(x - y) x 2 − y 2 = ( x + y ) ( x − y ) Cannot Factor
a 2 + b 2 is NOT factorable over reals a^2 + b^2 \text{ is NOT factorable over reals} a 2 + b 2 is NOT factorable over reals Perfect Square Trinomials Sum Pattern
a 2 + 2 a b + b 2 = ( a + b ) 2 a^2 + 2ab + b^2 = (a + b)^2 a 2 + 2 ab + b 2 = ( a + b ) 2 Difference Pattern
a 2 − 2 a b + b 2 = ( a − b ) 2 a^2 - 2ab + b^2 = (a - b)^2 a 2 − 2 ab + b 2 = ( a − b ) 2 Identification Test
First and last terms perfect squares, middle = ± 2 first ⋅ last \text{First and last terms perfect squares, middle = } \pm 2 \sqrt{\text{first} \cdot \text{last}} First and last terms perfect squares, middle = ± 2 first ⋅ last Example
x 2 + 6 x + 9 = ( x + 3 ) 2 x^2 + 6x + 9 = (x + 3)^2 x 2 + 6 x + 9 = ( x + 3 ) 2 Trinomials (Leading Coefficient = 1) Pattern
x 2 + b x + c = ( x + p ) ( x + q ) x^2 + bx + c = (x + p)(x + q) x 2 + b x + c = ( x + p ) ( x + q ) Find p and q
p + q = b , p ⋅ q = c p + q = b,\ p \cdot q = c p + q = b , p ⋅ q = c Example
x 2 + 5 x + 6 = ( x + 2 ) ( x + 3 ) x^2 + 5x + 6 = (x + 2)(x + 3) x 2 + 5 x + 6 = ( x + 2 ) ( x + 3 ) Negative Constant
x 2 + 5 x − 6 = ( x + 6 ) ( x − 1 ) x^2 + 5x - 6 = (x + 6)(x - 1) x 2 + 5 x − 6 = ( x + 6 ) ( x − 1 ) Trinomials (Leading Coefficient ≠ 1) Pattern
a x 2 + b x + c ax^2 + bx + c a x 2 + b x + c AC Method
Find factors of a c that sum to b \text{Find factors of } ac \text{ that sum to } b Find factors of a c that sum to b Example
2 x 2 + 7 x + 3 = ( 2 x + 1 ) ( x + 3 ) 2x^2 + 7x + 3 = (2x + 1)(x + 3) 2 x 2 + 7 x + 3 = ( 2 x + 1 ) ( x + 3 ) Reverse FOIL Check
( 2 x + 1 ) ( x + 3 ) = 2 x 2 + 6 x + x + 3 = 2 x 2 + 7 x + 3 ✓ (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3 \checkmark ( 2 x + 1 ) ( x + 3 ) = 2 x 2 + 6 x + x + 3 = 2 x 2 + 7 x + 3 ✓ Factoring by Grouping 4-Term Pattern
a x + a y + b x + b y = a ( x + y ) + b ( x + y ) = ( a + b ) ( x + y ) ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y) a x + a y + b x + b y = a ( x + y ) + b ( x + y ) = ( a + b ) ( x + y ) Example
x 3 + 2 x 2 + 3 x + 6 = x 2 ( x + 2 ) + 3 ( x + 2 ) = ( x 2 + 3 ) ( x + 2 ) x^3 + 2x^2 + 3x + 6 = x^2(x + 2) + 3(x + 2) = (x^2 + 3)(x + 2) x 3 + 2 x 2 + 3 x + 6 = x 2 ( x + 2 ) + 3 ( x + 2 ) = ( x 2 + 3 ) ( x + 2 ) Sum & Difference of Cubes Sum of Cubes
a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3 + b^3 = (a + b)(a^2 - ab + b^2) a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) Difference of Cubes
a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^3 - b^3 = (a - b)(a^2 + ab + b^2) a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) Mnemonic (SOAP)
Same, Opposite, Always Positive sign pattern \text{Same, Opposite, Always Positive sign pattern} Same, Opposite, Always Positive sign pattern Example (Sum)
x 3 + 8 = ( x + 2 ) ( x 2 − 2 x + 4 ) x^3 + 8 = (x + 2)(x^2 - 2x + 4) x 3 + 8 = ( x + 2 ) ( x 2 − 2 x + 4 ) Example (Difference)
27 − y 3 = ( 3 − y ) ( 9 + 3 y + y 2 ) 27 - y^3 = (3 - y)(9 + 3y + y^2) 27 − y 3 = ( 3 − y ) ( 9 + 3 y + y 2 ) Special Higher-Degree Patterns Difference of Fourth Powers
a 4 − b 4 = ( a 2 + b 2 ) ( a + b ) ( a − b ) a^4 - b^4 = (a^2 + b^2)(a + b)(a - b) a 4 − b 4 = ( a 2 + b 2 ) ( a + b ) ( a − b ) Trinomial as Quadratic-in-Form
x 4 + 5 x 2 + 6 = ( x 2 + 2 ) ( x 2 + 3 ) x^4 + 5 x^2 + 6 = (x^2 + 2)(x^2 + 3) x 4 + 5 x 2 + 6 = ( x 2 + 2 ) ( x 2 + 3 ) Sum/Difference of nth Powers
a n − b n = ( a − b ) ( a n − 1 + a n − 2 b + ⋯ + b n − 1 ) a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1}) a n − b n = ( a − b ) ( a n − 1 + a n − 2 b + ⋯ + b n − 1 ) Factoring Flowchart Step 1
Factor out GCF \text{Factor out GCF} Factor out GCF Step 2
Count terms: 2, 3, or 4? \text{Count terms: 2, 3, or 4?} Count terms: 2, 3, or 4? 2 Terms
Try diff of squares, sum/diff of cubes \text{Try diff of squares, sum/diff of cubes} Try diff of squares, sum/diff of cubes 3 Terms
Try perfect square, then trinomial methods \text{Try perfect square, then trinomial methods} Try perfect square, then trinomial methods 4 Terms
Try factoring by grouping \text{Try factoring by grouping} Try factoring by grouping Step 3
Check if any factor can be factored further \text{Check if any factor can be factored further} Check if any factor can be factored further 