Free GCSE Maths lesson: Algebra
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Lesson 29 · GCSE / Key Stage 4 · Maths · Algebra

Factorising Expressions

Learn how to put expressions back into brackets by finding common factors and checking by expanding.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

GCSE specification fit

A core algebra skill after expanding brackets.

Factorising rewrites an expression as a product. GCSE questions use it to simplify algebra, solve equations and prepare for quadratics.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandAlgebra
Tier guidanceFoundation and Higher

What you will learn

  • What factorising means.
  • How to find the highest common factor of algebraic terms.
  • How to factorise expressions into one bracket.
  • How to include variables in the common factor.
  • How to factorise simple quadratics into two brackets, including when a common factor comes out first.
  • How to check your answer by expanding again.

Why this matters

Factorising is the reverse of expanding. It makes expressions more useful when a question asks you to solve, simplify or spot a structure.

Good factorising also protects later marks in equation solving because the bracketed form often reveals the next step.

Prior knowledge

You should already be comfortable with:

  • finding common factors of numbers,
  • understanding terms such as 6x and 4x²,
  • expanding single brackets,
  • collecting like terms,
  • multiplying positive and negative numbers.

Clear explanation

What factorising means

To factorise, write an expression as something multiplied by a bracket. It is the reverse of expanding.

3(x + 4) expands to 3x + 12 3x + 12 factorises to 3(x + 4)

Use the highest common factor

Find the largest factor shared by all terms. Put it outside the bracket, then divide each term by that factor to fill the bracket.

6x + 18 The highest common factor of 6x and 18 is 6 6x + 18 = 6(x + 3)

Common factors can include letters

If every term contains the same letter factor, include it outside the bracket.

8a² − 12a The highest common factor is 4a 8a² − 12a = 4a(2a − 3)

Fully factorised means no common factor is left

GCSE questions often use the word fully. After taking out a factor, check whether the expression inside the bracket can still be factorised.

6x² + 18x = 6x(x + 3) The bracket x + 3 has no further common factor So 6x(x + 3) is fully factorised

When the first term is negative

You can take out a negative factor if it makes the bracket neater. Check the signs by expanding.

−5x − 20 = −5(x + 4) Check: −5 × x + −5 × 4 = −5x − 20

Simple quadratic factorising

Higher-tier questions may ask for two brackets. For x² + bx + c, look for two numbers that multiply to c and add to b.

x² + 7x + 12 Find two numbers that multiply to 12 and add to 7: 3 and 4 x² + 7x + 12 = (x + 3)(x + 4)

Check by expanding

The quickest check is to expand your factorised answer. If it returns to the original expression, the factorising works.

5(y − 2) = 5y − 10 So 5y − 10 = 5(y − 2)

Worked examples

Example 1: Factorise using a number

Factorise 4x + 16.

The highest common factor is 4. 4x + 16 = 4(x + 4)
Answer: 4(x + 4).

Example 2: Include the variable

Factorise 9m² + 6m.

The highest common factor is 3m. 9m² + 6m = 3m(3m + 2)
Answer: 3m(3m + 2).

Example 3: Factorise with subtraction

Factorise 15p − 25.

The highest common factor is 5. 15p − 25 = 5(3p − 5)
Answer: 5(3p − 5).

Example 4: Simple quadratic

Factorise x² + 9x + 20.

5 × 4 = 20 and 5 + 4 = 9. x² + 9x + 20 = (x + 5)(x + 4)
Answer: (x + 5)(x + 4).

Quick checks

Choose an answer, then check your thinking.

1. Factorise 5x + 10.

2. Factorise 6y² − 9y.

3. Factorise x² + 6x + 8.

Practice questions

Question 1

A rectangle has width 3 cm and length (x + 4) cm. Expand the area, then factorise your expression back into brackets.

Reveal answer and marking guidance

Answer: The expanded area is 3x + 12, and the factorised form is 3(x + 4).

Marking: Show that 3(x + 4) expands to 3x + 12, then use the common factor 3 to return to the bracket form.

Question 2

A pupil writes 8a − 20 = 4(2a + 5). Explain the mistake and give the correct factorised form.

Reveal answer and marking guidance

Answer: The sign inside the bracket should be negative, so 8a − 20 = 4(2a − 5).

Marking: The highest common factor is 4; expanding 4(2a − 5) gives 8a − 20, but 4(2a + 5) gives 8a + 20.

Question 3

Factorise 10y² + 15y fully, and explain why the letter y belongs outside the bracket.

Reveal answer and marking guidance

Answer: 5y(2y + 3).

Marking: Both terms contain 5 and y, so the highest common factor is 5y. Dividing gives 10y² ÷ 5y = 2y and 15y ÷ 5y = 3.

Question 4

Factorise −6m − 18 by taking out a negative common factor, then expand to check your answer.

Reveal answer and marking guidance

Answer: −6(m + 3), and expanding gives −6m − 18.

Marking: Taking out −6 gives a positive bracket. The check is −6 × m + −6 × 3 = −6m − 18.

Question 5

Factorise x² + 8x + 15.

Reveal answer and marking guidance

Answer: (x + 3)(x + 5).

Marking: 3 × 5 = 15 and 3 + 5 = 8.

Question 6

Factorise x² − x − 12.

Reveal answer and marking guidance

Answer: (x − 4)(x + 3).

Marking: −4 × 3 = −12 and −4 + 3 = −1.

Question 7

Factorise 12b² − 18b fully.

Reveal answer and marking guidance

Answer: 6b(2b − 3).

Marking: Use the full common factor 6b, then check that 6b × 2b = 12b² and 6b × −3 = −18b.

Question 8

Factorise 3x² + 18x + 24 fully.

Reveal answer and marking guidance

Answer: 3(x + 2)(x + 4).

Marking: Take out 3 first to get 3(x² + 6x + 8), then factorise x² + 6x + 8 as (x + 2)(x + 4).

Question 9

Factorise 4x² − 12x − 40 fully.

Reveal answer and marking guidance

Answer: 4(x − 5)(x + 2).

Marking: Take out 4 first to get 4(x² − 3x − 10), then use the pair −5 and +2 because −5 × 2 = −10 and −5 + 2 = −3.

Question 10

Factorise 6x² + 30x + 36 fully.

Reveal answer and marking guidance

Answer: 6(x + 2)(x + 3).

Marking: Take out 6 first to get 6(x² + 5x + 6), then factorise x² + 5x + 6 as (x + 2)(x + 3).

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For factorising, marks usually come from identifying the highest common factor, dividing each term correctly into the bracket, checking whether the expression is fully factorised, keeping signs with the terms they belong to, and expanding the final bracket form to check it.

Common mistakes

  • Not taking the highest common factor: 12x + 18 can be 6(2x + 3), not just 3(4x + 6).
  • Forgetting the letter factor: 10y² + 15y should include 5y outside the bracket.
  • Losing a negative sign: 8a − 20 becomes 4(2a − 5), not 4(2a + 5).
  • Leaving unmatched brackets: factorised answers must be written as a product such as 4(x + 2).
  • Missing a common factor before two brackets: 3x² + 18x + 24 should start by taking out 3.
  • Not checking quadratics: expand two brackets again to confirm the middle term.

Extension challenge

Factorise 2x² + 10x + 12 fully.

Reveal answer

Answer: 2(x + 2)(x + 3).

First take out 2 to get 2(x² + 5x + 6), then x² + 5x + 6 = (x + 2)(x + 3).

Exam-board guidance

Factorising expressions is a core GCSE algebra skill for every board. It links to expanding, solving equations, simplifying expressions and quadratic methods.

AQA GCSE Maths

Take out the full highest common factor, include any shared letter powers, and expand your bracket to check the original expression returns exactly before moving to equation work.

OCR GCSE Maths

Show the common factor first, keep signs attached to their terms inside the bracket, and use expansion as your accuracy check.

Pearson Edexcel GCSE Maths

For quadratics, check first for a common factor, then check that the two bracket numbers multiply to the constant and add to the x-coefficient.

Eduqas GCSE Maths

Use factorising as a reverse-expansion check, especially before solving, simplifying or showing that two expressions are equivalent.

WJEC Wales

Factorising may appear inside formula or problem-solving questions, so keep the bracketed expression linked to the context and preserve signs.

CCEA GCSE Maths

Factorising can be needed before solving, so show the full common factor and check negative signs carefully in both calculator and non-calculator units.

Next lesson

Next, move into Linear Equations.

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