Factoring Out the GCF Examples in Math

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

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

Concept Recap

Factoring out the greatest common factor (GCF) means identifying the largest expression that divides every term, then rewriting the polynomial as that GCF times what remains.

Look at what all terms share in common—like taking the common ingredient out of a recipe. In 6x3+9x26x^3 + 9x^2, every term has at least 3x23x^2 in it, so pull it out front: 3x2(2x+3)3x^2(2x + 3).

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 the GCF rewrites a polynomial as the largest common factor times what is left.

Common stuck point: The procedure for factoring out the gcf is the easy part; the trap is taking the highest variable power instead of the lowest. Asking "Do all terms share a numerical and/or variable factor I can lift to the front?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do all terms share a numerical and/or variable factor I can lift to the front?

Worked Examples

Example 1

easy
Factor 6x2+9x6x^2 + 9x.

Answer

3x(2x+3)3x(2x + 3)

First step

1
Step 1: Find the GCF of 6x26x^2 and 9x9x: GCF of 6 and 9 is 3; both have at least xx. GCF = 3x3x.

Full solution

  1. 2
    Step 2: Divide each term: 6x2÷3x=2x6x^2 \div 3x = 2x and 9x÷3x=39x \div 3x = 3.
  2. 3
    Step 3: Write as product: 3x(2x+3)3x(2x + 3).
  3. 4
    Check: 3x(2x+3)=6x2+9x3x(2x + 3) = 6x^2 + 9x ✓
Factoring out the GCF is the reverse of distribution. Find the largest factor common to every term, divide each term by it, and write the result as a product.

Example 2

medium
Factor 12x3−8x2+4x12x^3 - 8x^2 + 4x.

Example 3

medium
Factor: 12x2y−18xy212x^2y - 18xy^2.

Example 4

medium
Factor: 20a3b2−30a2b3+50ab20a^3b^2 - 30a^2b^3 + 50ab.

Example 5

medium
Factor: x(x+2)+3(x+2)x(x + 2) + 3(x + 2).

Example 6

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

Example 7

hard
Factor: 36x5y3−24x4y4+60x3y236x^5y^3 - 24x^4y^4 + 60x^3y^2.

Example 8

hard
Factor by grouping: x3+2x2+3x+6x^3 + 2x^2 + 3x + 6.

Example 9

hard
Factor: 12x2+34x\tfrac{1}{2}x^2 + \tfrac{3}{4}x.

Example 10

hard
Factor by grouping: 2x3−6x2+5x−152x^3 - 6x^2 + 5x - 15.

Example 11

challenge
Factor completely: 6x4−24x26x^4 - 24x^2.

Practice Problems

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

Example 1

easy
Factor 10x+1510x + 15.

Example 2

hard
Factor −18x4y2+12x3y3−6x2y-18x^4y^2 + 12x^3y^3 - 6x^2y.

Example 3

easy
Factor out the GCF: 2x+62x+6.

Example 4

easy
Factor out the GCF: 3x2+9x3x^2+9x.

Example 5

easy
Factor out the GCF: 4x2+8x4x^2+8x.

Example 6

easy
Factor out the GCF: 5x3+10x25x^3+10x^2.

Example 7

easy
Factor out the GCF: 6x+96x+9.

Example 8

easy
Factor out the GCF: x2+xx^2+x.

Example 9

easy
Factor out the GCF: 7x2−77x^2-7.

Example 10

easy
Factor out the GCF: 10x3−5x10x^3-5x.

Example 11

medium
Factor out the GCF: 12x3+18x2−6x12x^3+18x^2-6x.

Example 12

medium
Factor out the GCF: 8x2y+12xy28x^2y+12xy^2.

Example 13

medium
Factor out the GCF and verify: 15x4−25x215x^4-25x^2.

Example 14

medium
Factor: x(x+2)+3(x+2)x(x+2)+3(x+2) by taking out the common binomial.

Example 15

medium
Factor by grouping: x3+2x2+3x+6x^3+2x^2+3x+6.

Example 16

medium
Factor out the GCF: −2x2−4x-2x^2-4x.

Example 17

medium
Factor out the GCF: 9x2y3−6x3y2+3x2y29x^2y^3-6x^3y^2+3x^2y^2.

Example 18

medium
A box has volume 4x3+8x24x^3+8x^2. Factor to express it as height times base area, where height is the GCF's variable part.

Example 19

medium
Factor out the GCF: 14x4+21x3−7x214x^4+21x^3-7x^2.

Example 20

challenge
Factor completely: 6x4−6x26x^4-6x^2.

Example 21

challenge
Factor out the GCF including a binomial power: 4(x+1)3−8(x+1)24(x+1)^3-8(x+1)^2.

Example 22

challenge
For what value of kk is 3x3x the GCF of 6x2+kx6x^2+kx? Find kk and the factored form.

Example 23

easy
Factor out the GCF: 8x+128x + 12.

Example 24

easy
Factor: x3+x2x^3 + x^2.

Example 25

easy
Factor: 9x−39x - 3.

Example 26

easy
Factor: 6a2+9a6a^2 + 9a.

Example 27

easy
Factor: 15x2−25x15x^2 - 25x.

Example 28

medium
Factor: 24x4+16x3−8x224x^4 + 16x^3 - 8x^2.

Example 29

medium
Factor: −12x3+18x2-12x^3 + 18x^2.

Example 30

medium
Factor: 4x(2x−1)−7(2x−1)4x(2x - 1) - 7(2x - 1).

Example 31

medium
Factor: 5x4+10x3+5x25x^4 + 10x^3 + 5x^2.

Example 32

medium
Verify: does 3x(x+4)=3x2+12x3x(x + 4) = 3x^2 + 12x?

Example 33

hard
Factor completely: 2x3−8x2x^3 - 8x.

Example 34

hard
Factor: x(2x−3)+5(3−2x)x(2x - 3) + 5(3 - 2x).

Example 35

hard
Factor: 3xn+2+6xn+1−9xn3x^{n+2} + 6x^{n+1} - 9x^n for integer n≥0n \geq 0.
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Background Knowledge

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

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