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Algebra · Higher

Expanding & factorising

Expanding removes brackets by multiplying every term inside by the term outside. Factorising is the reverse — writing an expression as a product of factors. At Higher tier you need to expand double brackets and factorise quadratics.

Revising expanding & factorising the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Expanding & factorising GCSE” on the site for the complete guide.

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Key facts to remember

  • 1Expanding double brackets: (a + b)(c + d) = ac + ad + bc + bd (FOIL method).
  • 2(a + b)² = a² + 2ab + b² — a perfect square.
  • 3(a + b)(a − b) = a² − b² — difference of two squares.
  • 4To factorise x² + bx + c, find two numbers that multiply to c and add to b.
  • 5Always check your factorisation by expanding back out.
  • 6Take out common factors first before attempting to factorise quadratics.
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Formulas

FOIL expansion
(a + b)(c + d) = ac + ad + bc + bd
Perfect square
(a + b)² = a² + 2ab + b²
Difference of two squares
(a + b)(a − b) = a² − b²
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Worked examples

Example 1

Expand and simplify (2x + 3)(x − 4).

Working

  1. Multiply First terms: 2x × x = 2x²
  2. Multiply Outer terms: 2x × (−4) = −8x
  3. Multiply Inner terms: 3 × x = 3x
  4. Multiply Last terms: 3 × (−4) = −12
  5. Collect like terms: 2x² − 8x + 3x − 12 = 2x² − 5x − 12
Answer2x² − 5x − 12
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. Write as (x + 3)(x + 4).
  4. Check by expanding: x² + 4x + 3x + 12 = x² + 7x + 12 ✓
Answer(x + 3)(x + 4)
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Common mistakes

✗Forgetting the middle terms when expanding (only writing a² + b² instead of a² + 2ab + b²).
✗Getting signs wrong when one bracket contains a negative.
✗Not checking the factorisation by re-expanding.
✗Forgetting to take out a common factor first.
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Exam tips

✓Always show the four products from FOIL before collecting like terms.
✓For "factorise fully", check for a common factor first, then factorise the quadratic.
✓If asked to "expand and simplify", collect like terms in your final answer.
✓Practise the difference of two squares pattern — it comes up a lot.

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