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Learn Index · Patterns and Algebra Grade 9–10

Factoring algebraic expressions

Writing an expression as a product, the reverse of expanding.

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Factoring undoes multiplying

Factoring means writing an expression as a product.

  • Expanding: (x + 2)(x + 3) → x² + 5x + 6
  • Factoring: x² + 5x + 6 → (x + 2)(x + 3)

Factoring is essential for solving quadratic equations and simplifying rational expressions.

Step 1: Always look for a GCF

Take out the greatest common factor first.

  • 6x + 9 = 3(2x + 3)
  • 4x³ − 10x² = 2x²(2x − 5)
  • 5ab + 15a = 5a(b + 3)

Step 2: Factor what’s left

Pattern How to factor Example
x² + bx + c find two numbers that multiply to c and add to b x² + 7x + 12 = (x + 3)(x + 4)
ax² + bx + c (a ≠ 1) find numbers multiplying to ac and adding to b; split and group 2x² + 7x + 3 = (2x + 1)(x + 3)
a² − b² difference of squares x² − 49 = (x + 7)(x − 7)
a² ± 2ab + b² perfect square x² + 10x + 25 = (x + 5)²
four terms grouping see below

Simple trinomials: x² + bx + c

Example: x² − x − 12

Find two numbers that multiply to −12 and add to −1: −4 and 3.

→ (x − 4)(x + 3)

Sign tips:

c b Numbers are
positive positive both positive
positive negative both negative
negative either one positive, one negative

Complex trinomials: ax² + bx + c

Example: 3x² + 10x + 8

  1. ac = 24. Numbers that multiply to 24 and add to 10: 4 and 6.
  2. Split the middle: 3x² + 4x + 6x + 8
  3. Group: x(3x + 4) + 2(3x + 4)
  4. Factor: (3x + 4)(x + 2)

Grouping four terms

Example: xy + 3x + 2y + 6

  • x(y + 3) + 2(y + 3) = (x + 2)(y + 3)

Factor completely

Keep going until nothing more factors.

Example: 2x³ − 18x

  • GCF: 2x(x² − 9)
  • Difference of squares: 2x(x + 3)(x − 3)

Check by expanding

Multiply your factored form back out. You should get the original.

Worked examples

Example 1: x² + 9x + 20 → (x + 4)(x + 5).

Example 2: 16y² − 1 → (4y + 1)(4y − 1).

Example 3: 2x² − 5x − 3 → ac = −6: −6 and 1 → 2x² − 6x + x − 3 → 2x(x − 3) + 1(x − 3) → (2x + 1)(x − 3).

Common mistakes

  • Skipping the GCF, leaving the expression only partly factored.
  • Sign errors in the two numbers.
  • Trying to factor x² + 9 as (x + 3)(x + 3). A sum of squares doesn’t factor over the real numbers.

Try it yourself

  1. 8a − 12
  2. x² + 11x + 28
  3. x² − 3x − 40
  4. 9m² − 25
  5. 6x² + x − 2

Answers: 1) 4(2a − 3) 2) (x + 4)(x + 7) 3) (x − 8)(x + 5) 4) (3m + 5)(3m − 5) 5) (3x + 2)(2x − 1)

Quick recap

  • Always take out the greatest common factor first.
  • Recognize trinomials, difference of squares, perfect squares and grouping.
  • Check by expanding.