Factoring by Grouping Examples in Math

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

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

Concept Recap

A factoring technique for polynomials with four or more terms: group terms into pairs, factor the GCF from each pair, then factor out the common binomial factor.

Imagine four terms that seem unrelated. By cleverly grouping them into two pairs and factoring each pair separately, a common binomial factor often emerges—like finding a hidden pattern by rearranging puzzle pieces.

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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: Group four terms into two pairs so a common binomial pops out and can be factored once more.

Common stuck point: The procedure for factoring by grouping is the easy part; the trap is sign error pulling the GCF from the second pair. Asking "After factoring the GCF from each pair, do both pairs leave behind the identical binomial?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: After factoring the GCF from each pair, do both pairs leave behind the identical binomial?

Worked Examples

Example 1

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

Answer

(x2+2)(x+3)(x^2 + 2)(x + 3)

First step

1
Step 1: Group into pairs: (x3+3x2)+(2x+6)(x^3 + 3x^2) + (2x + 6).

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Example 2

hard
Factor 6x3−9x2−4x+66x^3 - 9x^2 - 4x + 6 by grouping.

Example 3

medium
Factor 3x3−6x2−5x+103x^3 - 6x^2 - 5x + 10 by grouping.

Example 4

medium
Factor 6x2+11x+36x^2 + 11x + 3 by AC-method grouping.

Example 5

hard
Factor x3+2x2−4x−8x^3 + 2x^2 - 4x - 8 completely.

Example 6

hard
Use grouping to solve x3−2x2−9x+18=0x^3 - 2x^2 - 9x + 18 = 0.

Example 7

challenge
Show that x3+3x2−x−3x^3 + 3x^2 - x - 3 factors by grouping, and find all real roots.

Practice Problems

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

Example 1

easy
Factor xy+3x+2y+6xy + 3x + 2y + 6.

Example 2

medium
Factor 2x3−x2+8x−42x^3 - x^2 + 8x - 4.

Example 3

easy
Factor x3+x2+x+1x^3 + x^2 + x + 1 by grouping.

Example 4

easy
Factor xy+3y+2x+6xy + 3y + 2x + 6 by grouping.

Example 5

easy
Factor 2x3+6x2+x+32x^3 + 6x^2 + x + 3 by grouping.

Example 6

easy
Factor ab+4a+2b+8ab + 4a + 2b + 8 by grouping.

Example 7

easy
Factor x3−2x2+5x−10x^3 - 2x^2 + 5x - 10 by grouping.

Example 8

easy
Factor 3x3+12x2+2x+83x^3 + 12x^2 + 2x + 8 by grouping.

Example 9

easy
Factor 5x+5y+ax+ay5x + 5y + ax + ay by grouping.

Example 10

easy
Factor x3+3x2+4x+12x^3 + 3x^2 + 4x + 12 by grouping.

Example 11

medium
Factor 6x3−9x2−4x+66x^3 - 9x^2 - 4x + 6 by grouping.

Example 12

medium
Factor 2x3+x2−6x−32x^3 + x^2 - 6x - 3 by grouping.

Example 13

medium
Factor 3x2+7x+23x^2 + 7x + 2 using grouping (AC method).

Example 14

medium
Factor 4x3+8x2+3x+64x^3 + 8x^2 + 3x + 6 by grouping.

Example 15

medium
Factor x3+2x2−9x−18x^3 + 2x^2 - 9x - 18 completely by grouping.

Example 16

medium
Factor 2ax−4ay+3bx−6by2ax - 4ay + 3bx - 6by by grouping.

Example 17

medium
Factor x3−x2−x+1x^3 - x^2 - x + 1 completely by grouping.

Example 18

medium
Factor x3+4x2+3x+12x^3 + 4x^2 + 3x + 12 by grouping.

Example 19

medium
Factor 10x3−15x2−4x+610x^3 - 15x^2 - 4x + 6 by grouping.

Example 20

challenge
Factor x3−3x2−4x+12x^3 - 3x^2 - 4x + 12 completely.

Example 21

challenge
Factor 6x3+4x2−9x−66x^3 + 4x^2 - 9x - 6 by grouping.

Example 22

challenge
Factor x3+5x2−2x−10x^3 + 5x^2 - 2x - 10 and identify all real roots.

Example 23

easy
Factor x3+4x2+3x+12x^3 + 4x^2 + 3x + 12 by grouping.

Example 24

easy
Factor x3−5x2+2x−10x^3 - 5x^2 + 2x - 10 by grouping.

Example 25

easy
Factor 2x3+10x2+3x+152x^3 + 10x^2 + 3x + 15 by grouping.

Example 26

medium
Factor 4x3+8x2−3x−64x^3 + 8x^2 - 3x - 6 by grouping.

Example 27

medium
Factor x3−7x2−4x+28x^3 - 7x^2 - 4x + 28 by grouping.

Example 28

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

Example 29

medium
Factor 4x2+8x+34x^2 + 8x + 3 by the AC method.

Example 30

medium
Factor x3+2x2−9x−18x^3 + 2x^2 - 9x - 18 by grouping and complete the factoring.

Example 31

medium
Factor 3y3+6y2+4y+83y^3 + 6y^2 + 4y + 8 by grouping.

Example 32

medium
Factor mn+5m−3n−15mn + 5m - 3n - 15 by grouping.

Example 33

hard
Factor 6x3+9x2−4x−66x^3 + 9x^2 - 4x - 6 by grouping.

Example 34

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

Example 35

hard
Factor 8x3+12x2+10x+158x^3 + 12x^2 + 10x + 15 by grouping.

Example 36

hard
Use grouping (AC method) to factor 6x2−13x+66x^2 - 13x + 6.

Example 37

hard
Factor x3+5x2−x−5x^3 + 5x^2 - x - 5 completely.

Example 38

medium
Factor 2ab+6a+b+32ab + 6a + b + 3 by grouping.

Example 39

hard
Factor 4x3−12x2+9x−274x^3 - 12x^2 + 9x - 27 by grouping.

Example 40

challenge
Factor x3+x2y−xy2−y3x^3 + x^2y - xy^2 - y^3 by grouping.
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Background Knowledge

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

factoring gcfpolynomial addition subtraction