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Edexcel · GCSE Maths · 1MA1 · Foundation and Higher

M8 · Expanding and factorising

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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.

    The zero-product rule says that if two factors multiply to zero, at least one factor is zero. Thus (x − 2)(x + 3) = 0 gives x − 2 = 0 or x + 3 = 0, so x = 2 or x = −3.

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.

    The leading coefficient is the number multiplying x2. For 6x2 + 7x + 2, find two numbers that multiply to 6 × 2 = 12 and add to 7: 3 and 4. This explains the split into 3x + 4x in the next example.

  • 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.

Mind map

Use the branches to recall the ideas and explain their connections. Check the revision notes for the full detail.

View M8 mind map
M8 M8 mind map: Expand, Common factors, Quadratics, Further products. A text version follows.
Open the full-size map to zoom. Use the printable view for A4 landscape or enlarge to A3.

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Read the mind map as text

Expand

  • Distribute: Every inside term; outside minus changes every sign
  • Two brackets: Every term × every term; collect middle terms

Common factors

  • Reverse: 12x²+8x = 4x(3x+2); expand to check
  • First step: Higher: 2x²−18 = 2(x−3)(x+3)

Quadratics

  • Sum / product: x²+7x+12 = (x+3)(x+4); check signs
  • Squares / solve: a²−b² = (a−b)(a+b); solving needs equation = 0

Further products

  • Non-monic: Higher: 6x²+7x+2 = (3x+2)(2x+1); check middle term
  • Three brackets: Higher: Expand two first, simplify, then multiply third

Connections

  • Expand → Common factors: Factorising reverses distribution
  • Expand → Quadratics: Expanded coefficients check quadratic factors

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