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Edexcel · GCSE Maths · 1MA1 · Foundation and Higher

M8 · Expanding and factorising

Revision notes, worked examples and methods for expanding and factorising.

Notes and quizzes ready · 50 questions · Video coming soon.

Revise the key ideas

Expanding brackets

  • Distribute the term outside a bracket to every term inside: 3(x + 4) = 3x + 12. A minus sign outside changes every sign: −(2x − 5) = −2x + 5.
  • Worked example: 4(2x − 3) − 2(x + 5) = 8x − 12 − 2x − 10 = 6x − 22. Expand before collecting like terms.
  • For two brackets, multiply every term in one by every term in the other. (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6.
    Area model for expanding two bracketsA symbolic rectangle with side lengths x plus three and x plus two is divided into x squared, three x, two x and six.x3x2x²3x2x6(x + 3)(x + 2) = x² + 5x + 6
    Area model for expanding two brackets
    Enlarge diagram: Area model for expanding two brackets
  • Useful identities are (a + b)2 = a2 + 2ab + b2 and (a − b)2 = a2 − 2ab + b2. The middle term is essential.

Factorising

  • Factorising reverses expanding. Take out the highest common factor: 12x2 + 8x = 4x(3x + 2). Expanding again checks both terms.
  • To factorise x2 + bx + c, find two numbers whose sum is b and product is c. x2 + 7x + 12 = (x + 3)(x + 4).
  • Worked example: x2 − x − 12 = (x − 4)(x + 3), because −4 + 3 = −1 and (−4) × 3 = −12.
  • A difference of two squares is a2 − b2 = (a − b)(a + b). Thus x2 − 25 = (x − 5)(x + 5). A sum of squares does not use this identity.
  • Factorising an expression does not by itself find x. To solve an equation, first make one side zero, then use the zero-product rule.

Higher — more complex products and quadratics

  • For ax2 + bx + c with a ≠ 1, find a factorisation that produces both the correct leading coefficient and middle term.
  • Worked example: 6x2 + 7x + 2 = 6x2 + 3x + 4x + 2 = 3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1).
  • For three brackets, expand two, simplify, then multiply the result by the third. (x + 1)(x − 1)(x + 2) = (x2 − 1)(x + 2) = x3 + 2x2 − x − 2.
  • Look for a common factor before a quadratic pattern: 2x2 − 18 = 2(x2 − 9) = 2(x − 3)(x + 3).

Test yourself

50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.

Revision video

Video coming soon.