Algebra · Quick Reference

Factoring Cheat Sheet

Algebra I · Lesson 7/17

In one line: factoring reverses multiplication — breaking a polynomial back down into the simpler expressions that were multiplied together to create it, which is essential for solving equations.

Key Ideas

1Greatest Common Factor (GCF) Factoring. Find the largest factor shared by every term, then divide it out: 6x^2 + 9x = 3x(2x + 3).

2Factoring Trinomials (a=1). For x^2 + bx + c, find two numbers that multiply to c and add to b; those numbers become the factors: x^2+5x+6 = (x+2)(x+3).

3Difference of Squares. a^2 - b^2 factors as (a+b)(a-b) — recognizable whenever you see two perfect squares separated by subtraction.

4Perfect Square Trinomials. a^2 + 2ab + b^2 factors as (a+b)^2, and a^2 - 2ab + b^2 factors as (a-b)^2.

5Checking Your Factoring. Multiply the factors back out (using FOIL or distribution) to confirm you get the original polynomial.

Worked Examples

Factor 4x^2 + 12x using the GCF.

4x(x + 3)

Factor x^2 + 7x + 12.

(x + 3)(x + 4)

Factor x^2 - 16 using the difference of squares.

(x + 4)(x - 4)

Formulas

Difference of squares: a^2 - b^2 = (a+b)(a-b)

Perfect square trinomial: a^2 + 2ab + b^2 = (a+b)^2

Practice Yourself

Factor 3x^2 + 6x using the GCF.

3x(x+2)

Factor x^2 + 5x + 6.

(x+2)(x+3)

Factor x^2 - 9.

(x+3)(x-3)