Factorise expressions

Objective. Take out the highest common numerical or algebraic factor.

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Learn: the key idea

Factorising expressions is the reverse of expanding brackets, taking out the highest common factor shared by every term and writing it outside a bracket. To factorise 6x + 9, find the highest common factor of 6 and 9, which is 3, giving 3(2x + 3). With algebraic terms like 4x² + 8x, both the number and the letter can be factored out, giving the highest common factor 4x, so the expression becomes 4x(x + 2). Always check by expanding the bracket back out to confirm it matches the original expression exactly.

Worked example

Factorise the expression 10x² + 15x completely.

1 $Find the highest common factor of 10x² and 15x: 5x$
2 $Divide each term by 5x: 10x² \div 5x = 2x and 15x \div 5x = 3$
3 $Write the factorised form: 5x(2x + 3)$
4 $Check by expanding: 5x \times 2x + 5x \times 3 = 10x² + 15x$

Answer: 5x(2x + 3)

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Common mistake. Pupils sometimes take out a factor that is not the highest common factor, factorising 8x + 12 as 2(4x + 6) instead of fully factorising using 4 to get 4(2x + 3), leaving a bracket that could be simplified further.

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