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Algebra · Foundation & Higher

Factorising

Factorising is the reverse of expanding — you write an expression as a product of factors. At Foundation level this includes taking out a common factor and factorising simple quadratics.

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Key facts to remember

  • 1To factorise, find the highest common factor (HCF) of all terms and place it outside brackets.
  • 2Check by expanding: you should get back to the original expression.
  • 3Factorising quadratics: x² + bx + c = (x + p)(x + q) where p + q = b and p × q = c.
  • 4Always look for a common factor before factorising a quadratic.
  • 5A fully factorised expression cannot be factorised further.
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Worked examples

Example 1

Factorise 6x² − 9x

Working

  1. HCF of 6x² and 9x is 3x
  2. = 3x(2x − 3)
  3. Check: 3x × 2x = 6x², 3x × (−3) = −9x ✓
Answer3x(2x − 3)
Example 2

Factorise x² + 7x + 12

Working

  1. Find two numbers that multiply to 12 and add to 7
  2. 3 × 4 = 12 and 3 + 4 = 7 ✓
  3. = (x + 3)(x + 4)
Answer(x + 3)(x + 4)
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Common mistakes

✗Not taking out the full HCF — leaving a further common factor inside the bracket.
✗Getting signs wrong in quadratic factorisation (e.g. (x + 3)(x − 4) when both should be positive).
✗Forgetting to check by expanding back.
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Exam tips

✓For quadratics: list factor pairs of c and check which pair sums to b.
✓Always check your factorisation by expanding — it only takes a few seconds.

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