Algebra visual reference

Factoring Patterns Cheat Sheet

Factoring reverses multiplication. Always look for a greatest common factor first, then compare the remaining polynomial with a recognizable pattern and multiply back to verify.

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How to read it

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A printable factoring patterns cheat sheet covering greatest common factors, difference of squares, perfect-square trinomials, general trinomials, and cubes. Read the label first, identify the matching structure in your problem, then confirm that any stated conditions apply before substituting values.

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Remove every factor shared by all terms.

Greatest common factor

\displaystyle ab+ac=a(b+c)

Example: $\displaystyle 6x^3+9x^2=3x^2(2x+3)$

Special products

These patterns factor immediately when every condition matches.

Difference of squares

\displaystyle a^2-b^2=(a-b)(a+b)

Example: $\displaystyle x^2-25=(x-5)(x+5)$

Perfect-square sum

\displaystyle a^2+2ab+b^2=(a+b)^2

Perfect-square difference

\displaystyle a^2-2ab+b^2=(a-b)^2

Sum of cubes

\displaystyle a^3+b^3=(a+b)(a^2-ab+b^2)

Difference of cubes

\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)

Trinomials

Use multiplication to verify the middle term.

Monic trinomial

\displaystyle x^2+bx+c=(x+m)(x+n)

Example: $\displaystyle x^2+7x+12=(x+3)(x+4)$

General trinomial

\displaystyle ax^2+bx+c=(px+q)(rx+s)

Example: $\displaystyle 6x^2+7x+2=(3x+2)(2x+1)$

How to remember it

GCF first, pattern second, multiply to check

1A difference of squares must have subtraction and two perfect squares.
2Perfect-square trinomials have matching outer squares and twice their product in the middle.
3For ax²+bx+c, the inner and outer products must combine to b.