Factoring Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factoring.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Rewriting an algebraic expression as a product of two or more simpler expressions that multiply to give the original.

Reverse distribution: instead of expanding (x+2)(x+3)(x+2)(x+3), you compress x2+5x+6x^2 + 5x + 6 into the same product.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Factoring rewrites an expression as a product of simpler factors that multiply back to the original.

Common stuck point: The procedure for factoring is the easy part; the trap is forgetting to pull out the greatest common factor first. Asking "Am I rewriting an expression as a product of simpler factors that multiply back to it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I rewriting an expression as a product of simpler factors that multiply back to it?

Worked Examples

Example 1

easy
Factor x2+7x+12x^2 + 7x + 12.

Answer

(x+3)(x+4)(x + 3)(x + 4)

First step

1
Find two numbers that multiply to 12 and add to 7: those are 3 and 4.

Full solution

  1. 2
    Write the factored form: (x+3)(x+4)(x + 3)(x + 4).
  2. 3
    Check by expanding: x2+4x+3x+12=x2+7x+12x^2 + 4x + 3x + 12 = x^2 + 7x + 12 ✓
To factor x2+bx+cx^2 + bx + c, find two numbers pp and qq such that p+q=bp + q = b and p⋅q=cp \cdot q = c. Then the factorization is (x+p)(x+q)(x + p)(x + q).

Example 2

medium
Factor 6x2+11x+36x^2 + 11x + 3.

Example 3

medium
Factor: x2+2x−15x^2 + 2x - 15.

Example 4

medium
Factor: 3x2−12x+93x^2 - 12x + 9.

Example 5

hard
Factor: x3+2x2−x−2x^3 + 2x^2 - x - 2 by grouping.

Example 6

medium
Factor completely: 2x2−82x^2 - 8.

Example 7

challenge
Factor: x3−8x^3 - 8 (difference of cubes).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Factor x2−16x^2 - 16.

Example 2

medium
Factor 3x2−12x3x^2 - 12x.

Example 3

easy
Factor: 6x+96x + 9.

Example 4

easy
Factor: x2−9x^2 - 9.

Example 5

easy
Factor: x2+5x+6x^2 + 5x + 6.

Example 6

easy
Factor: x2−7x+12x^2 - 7x + 12.

Example 7

easy
Factor: x2+x−12x^2 + x - 12.

Example 8

easy
Factor: 2x2+4x2x^2 + 4x.

Example 9

easy
Factor: x2+6x+9x^2 + 6x + 9.

Example 10

easy
Factor: 4y2−254y^2 - 25.

Example 11

medium
Factor: 2x2+7x+32x^2 + 7x + 3.

Example 12

medium
Factor: 3x2−10x+33x^2 - 10x + 3.

Example 13

medium
Factor: x3+2x2−x−2x^3 + 2x^2 - x - 2.

Example 14

medium
Factor: x2−16x^2 - 16.

Example 15

medium
Factor completely: 3x2−273x^2 - 27.

Example 16

medium
Factor: x2+4x+4x^2 + 4x + 4.

Example 17

medium
Solve x2−5x+6=0x^2 - 5x + 6 = 0 by factoring.

Example 18

medium
Factor: 6x2+11x−106x^2 + 11x - 10.

Example 19

medium
Factor: a3−8a^3 - 8.

Example 20

challenge
Factor: x4−5x2+4x^4 - 5x^2 + 4.

Example 21

challenge
Factor: x2+6x+9−y2x^2 + 6x + 9 - y^2.

Example 22

challenge
Factor: x2+5x−14x^2 + 5x - 14 using Simon's Favorite Factoring Trick.

Example 23

easy
Factor: 8x+128x + 12.

Example 24

easy
Factor: x2−25x^2 - 25.

Example 25

easy
Factor: x2+8x+15x^2 + 8x + 15.

Example 26

easy
Factor: x2−10x+16x^2 - 10x + 16.

Example 27

medium
Factor: 4x2−164x^2 - 16.

Example 28

medium
Factor: x2−8x+16x^2 - 8x + 16.

Example 29

medium
Factor: 2x2+7x+32x^2 + 7x + 3.

Example 30

medium
Factor: 9−x29 - x^2.

Example 31

medium
Factor: x2−5x−14x^2 - 5x - 14.

Example 32

hard
Factor: 6x2+x−26x^2 + x - 2.

Example 33

hard
Factor: x2+4x+4−y2x^2 + 4x + 4 - y^2.

Example 34

hard
Factor: x4−16x^4 - 16.

Example 35

easy
Factor: 5x2−10x5x^2 - 10x.

Example 36

medium
Factor: x2−4x−21x^2 - 4x - 21.

Example 37

hard
Factor: 3x2−11x−43x^2 - 11x - 4.

Example 38

medium
Factor: 9x2−19x^2 - 1.

Example 39

hard
Factor: x2+6xy+9y2x^2 + 6xy + 9y^2.

Example 40

hard
Factor: x2−12x+36x^2 - 12x + 36.

Example 41

hard
Factor: 5x2+15x+105x^2 + 15x + 10.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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