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GCSE Maths → Algebra → Factorising

Factorising GCSE Maths: Step-by-Step Guide

Factorising is one of the key algebra skills in GCSE maths. It means putting an expression back into brackets. In simple terms, factorising is the reverse of expanding brackets.

This topic is important because it appears in simplifying algebra, solving equations, quadratic equations, graphs, and exam-style problem solving. If a student is weak at factorising, many later GCSE algebra topics become harder.

Video explanation

A short Worthing Maths Tutor video explanation for factorising GCSE maths can be embedded here later to improve student engagement and time on page.

What does factorising mean?

Factorising means finding a common factor and placing it outside a bracket.

3x + 12 = 3(x + 4)

Here, 3 is the common factor because both 3x and 12 can be divided by 3.

Exam tip: Always check your answer by expanding the bracket again. If you get the original expression, your factorising is correct.

Why students struggle with factorising

Many GCSE students understand factorising when they watch an example, but struggle when they have to do it alone. This usually happens because they are unsure how to spot the highest common factor.

  • They only factorise the numbers and forget the letters.
  • They take out a factor that is not common to every term.
  • They make sign mistakes with negative terms.
  • They confuse expanding with factorising.
  • They do not check by expanding back.

Method: factorising by taking out a common factor

  1. Look at every term in the expression.
  2. Find the biggest number that divides into all terms.
  3. Check whether any letters are also common to all terms.
  4. Put the common factor outside the bracket.
  5. Divide each term by the common factor to fill the bracket.
  6. Expand the bracket to check your answer.

Example 1: Factorise 6x + 18

The highest common factor of 6x and 18 is 6.

Put 6 outside the bracket:

6x + 18 = 6(x + 3)

Check: 6 × x = 6x and 6 × 3 = 18.

Example 2: Factorise 8a - 20

The highest common factor of 8a and 20 is 4.

8a - 20 = 4(2a - 5)

The minus sign stays inside the bracket because the second term is negative.

Common mistake: A common wrong answer is 4(2a + 5). This is incorrect because expanding it gives 8a + 20, not 8a - 20.

Factorising when letters are also common

Sometimes the common factor includes both a number and a letter.

Example 3: Factorise 6x² + 9x

The number part has a common factor of 3.

Both terms also contain x.

So the highest common factor is 3x.

6x² + 9x = 3x(2x + 3)

Check: 3x × 2x = 6x² and 3x × 3 = 9x.

Example 4: Factorise 12y² - 8y

The highest common factor of 12y² and 8y is 4y.

12y² - 8y = 4y(3y - 2)

Factorising simple quadratics

GCSE Higher students also need to factorise quadratics. These are expressions with x², such as:

x² + 7x + 12

To factorise this, find two numbers that multiply to make 12 and add to make 7.

Example 5: Factorise x² + 7x + 12

We need two numbers that multiply to 12 and add to 7.

The numbers are 3 and 4.

x² + 7x + 12 = (x + 3)(x + 4)

Example 6: Factorise x² - 5x + 6

We need two numbers that multiply to 6 and add to -5.

The numbers are -2 and -3.

x² - 5x + 6 = (x - 2)(x - 3)
Exam tip: For quadratic factorising, the signs matter. If the final number is positive and the middle term is negative, both brackets usually contain negative signs.

Common mistakes in GCSE factorising

  • Taking out 2 when a bigger common factor is possible.
  • Forgetting to include x as part of the common factor.
  • Changing signs incorrectly inside the bracket.
  • Leaving the answer only partly factorised.
  • Not checking the answer by expanding.

Practice questions

Try these without looking at the answers first.

  1. Factorise 4x + 20
  2. Factorise 7a - 21
  3. Factorise 10y + 15
  4. Factorise 8x² + 12x
  5. Factorise 18m² - 12m
  6. Factorise x² + 9x + 20
  7. Factorise x² + 8x + 15
  8. Factorise x² - 7x + 10

Answers

  1. 4(x + 5)
  2. 7(a - 3)
  3. 5(2y + 3)
  4. 4x(2x + 3)
  5. 6m(3m - 2)
  6. (x + 4)(x + 5)
  7. (x + 3)(x + 5)
  8. (x - 5)(x - 2)

Factorising FAQ

What does factorising mean in GCSE maths?

Factorising means rewriting an expression by putting it into brackets. It is the reverse of expanding brackets.

How do you check if factorising is correct?

You can check factorising by expanding the brackets again. If you get back to the original expression, the factorising is correct.

Why do students struggle with factorising?

Students often struggle because they miss the highest common factor, forget variable factors, make sign mistakes, or confuse factorising with expanding.

Is factorising needed for GCSE Higher maths?

Yes. Factorising is important for GCSE Higher maths, especially when solving quadratic equations and simplifying algebraic expressions.

Related GCSE algebra topics

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