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MathidiaMathematics Academy
Advanced algebraic manipulation
Lesson 12 of 75
Easy practice

Factorising complex expressions

Objective. Identify common, quadratic and difference-of-squares structures.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Factorising complex expressions combines several strategies: taking out a common factor first, then looking for a difference of two squares (a² − b² = (a + b)(a − b)) or a quadratic trinomial that splits into two brackets. For example, 2x² − 8 first has common factor 2, giving 2(x² − 4), which is then a difference of squares: 2(x − 2)(x + 2). Always check for a common factor before attempting any other method, since it simplifies the remaining expression considerably.

Worked example

Factorise the expression 3x² − 12 completely, identifying any common factor and difference of squares.

Diagram of a square
  1. 1Take out the common factor 3: 3(x² − 4).
  2. 2Recognise x² − 4 as a difference of two squares.
  3. 3Factorise: x² − 4 = (x − 2)(x + 2).
  4. 4Combine fully: 3(x − 2)(x + 2).

Answer: 3(x − 2)(x + 2)

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Common mistake. Pupils jump straight to difference-of-squares rules without removing a common factor first, leaving the answer only partly factorised, such as writing x² − 4 inside brackets still multiplied by 2 incorrectly. Always extract common factors first.

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