AlgebraFoundation & HigherTopic 13 of 73

Factorising Expressions –

GCSEMathsAI Team·8 min read·18 March 2026

Factorising Expressions

Factorising is one of the most important algebra skills you will need for your GCSE Maths exam. It appears on both Foundation and Higher tier papers across AQA, Edexcel, and OCR, and it underpins many other topics such as solving quadratic equations, simplifying algebraic fractions, and sketching graphs. On this page you will learn what factorising means, how to factorise single brackets and double brackets, and how to avoid the mistakes that cost students marks every year. Whether you are aiming for a grade 4 or a grade 9, mastering factorising will give you a significant advantage.

What Is Factorising?

Factorising is the reverse of expanding brackets. When you expand, you multiply out; when you factorise, you put an expression back into brackets by finding common factors.

Think of it as "un-doing" multiplication. If expanding 3(x + 4) gives 3x + 12, then factorising 3x + 12 gives 3(x + 4).

There are three main types of factorising at GCSE:

  1. Single bracket (common factor) factorising — take out the highest common factor (HCF) of all terms.
  2. Double bracket factorising — rewrite a quadratic expression as the product of two linear brackets.
  3. Difference of two squares — a special pattern where a² − b² = (a + b)(a − b).

Key formulas to remember:

ab + ac = a(b + c)
x² + (a+b)x + ab = (x + a)(x + b)
a² − b² = (a + b)(a − b)

Step-by-Step Method

Single Bracket Factorising

  1. Identify every term in the expression. For example, in 6x² + 9x there are two terms: 6x² and 9x.
  2. Find the HCF of the coefficients (numbers). The HCF of 6 and 9 is 3.
  3. Find the HCF of the variables. Both terms contain at least one x, so x is a common factor.
  4. Write the HCF outside the bracket. That gives 3x( ).
  5. Divide each term by the HCF and write the results inside the bracket: 3x(2x + 3).
  6. Check by expanding. 3x × 2x = 6x² and 3x × 3 = 9x. Correct.

Double Bracket Factorising (monic quadratics, where coefficient of x² is 1)

  1. Write the expression in the form x² + bx + c.
  2. Find two numbers that multiply to give c and add to give b.
  3. Place those numbers in two brackets: (x + p)(x + q).
  4. Expand to check.

Difference of Two Squares

  1. Confirm the expression has two terms, both perfect squares, separated by a minus sign.
  2. Take the square root of each term.
  3. Write as (√first + √second)(√first − √second).

Worked Example 1 — Foundation Level

Question: Factorise fully 12x²y + 18xy².

Working:

  • Coefficients: HCF of 12 and 18 = 6.
  • Variables: both terms contain at least one x and one y, so xy is common.
  • HCF overall = 6xy.
  • Divide each term: 12x²y ÷ 6xy = 2x; 18xy² ÷ 6xy = 3y.
  • Write the answer: 6xy(2x + 3y).

Check: 6xy × 2x = 12x²y ✓ and 6xy × 3y = 18xy² ✓

Answer: 6xy(2x + 3y)

Worked Example 2 — Higher Level

Question: Factorise x² − 5x − 14.

Working:

We need two numbers that multiply to −14 and add to −5.

Consider factor pairs of −14: (1, −14), (−1, 14), (2, −7), (−2, 7).

The pair (2, −7) gives 2 + (−7) = −5. That works.

So x² − 5x − 14 = (x + 2)(x − 7).

Check: x × x = x², x × (−7) = −7x, 2 × x = 2x, 2 × (−7) = −14. Combine: x² − 5x − 14 ✓

Answer: (x + 2)(x − 7)

Common Mistakes

  • Not fully factorising. Students write 3(4x + 6) instead of taking out the full HCF to get 6(2x + 3). Always check whether the terms inside the bracket still share a common factor.
  • Forgetting to factorise the variable part. In 8x³ + 12x², the HCF is 4x², not just 4. Look at the lowest power of each variable present in every term.
  • Sign errors in double brackets. When c is negative, one of your two numbers must be negative and one positive. Write out all factor pairs carefully.
  • Mixing up factorising and expanding. The question says "factorise" but some students expand instead. Read the instruction twice.
  • Ignoring difference of two squares. If you see something like 25 − 4x², recognise it as (5 + 2x)(5 − 2x) rather than trying to use double brackets with an x² term.

Exam Tips

  1. "Factorise fully" means take out the complete HCF. If the examiner writes "fully," they expect every common factor removed. You will lose a mark if anything common remains inside the bracket.
  2. Always expand to check. It takes 20 seconds and guarantees your answer is correct. In a high-stakes exam, that is time well spent.
  3. Look for difference of two squares first. It is quicker than double brackets and students often miss it. Any expression of the form a² − b² fits this pattern.
  4. On AQA and Edexcel, factorising is often the first step in a multi-part question — for instance, factorise then solve. Getting this step right unlocks the remaining marks.

Practice Questions

Q1 (Foundation): Factorise 15x + 25.

Answer: 5(3x + 5)

Q2 (Foundation/Higher): Factorise x² + 7x + 12.

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

Q3 (Higher): Factorise 49y² − 16.

Answer: (7y + 4)(7y − 4)

Ready to practise more factorising questions with instant feedback? Start revising with GCSEMathsAI — our AI tutor adapts to your level and helps you build confidence before exam day.

For a full list of every GCSE Maths topic, see our GCSE Maths Topics Complete List.

Summary

  • Factorising is the reverse of expanding brackets.
  • Single bracket factorising involves taking out the highest common factor (HCF) of all terms.
  • Double bracket factorising rewrites a quadratic as the product of two linear expressions.
  • Difference of two squares uses the pattern a² − b² = (a + b)(a − b).
  • Always factorise fully — check that nothing common remains inside the bracket.
  • Expand your answer to verify it matches the original expression.
  • Factorising is a gateway skill for solving quadratics, simplifying fractions, and many other Higher topics.
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