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Factorising Expressions

Factorising is expanding in reverse: pull the common factor back out of every term.

Factorising Is Reverse Expanding

To factorise 4 x + 12, ask: what is the biggest number that divides into both 4 x and 12 ? It is 4 . Divide each term by 4, and write the results inside a bracket, with the 4 outside. 4 x \div 4 = x, and 12 \div 4 = 3, so 4 x + 12 = 4 (x + 3). You can always check a factorising answer by expanding it back out — if you get the original expression, it is correct.

Example 1 — easy numbers

Factorise 3 x + 9 . 1 Find the highest common factor of 3 x and 9 . It is 3 . 2 Divide each term by 3 : 3 x \div 3 = x, and 9 \div 3 = 3 . 3 Write the result: 3 (x + 3) 4 Check by expanding: 3 (x + 3) = 3 x + 9 . ✓ Answer: 3 (x + 3)

Example 2 — a common letter

Factorise 5 x^{2} + 10 x . 1 Both terms contain x, so x is a common factor. Both terms are also multiples of 5 . 2 The highest common factor is 5 x . 3 Divide each term by 5 x : 5 x^{2} \div 5 x = x, and 10 x \div 5 x = 2 . 4 Write the result: 5 x (x + 2) 5 Check by expanding: 5 x (x + 2) = 5 x^{2} + 10 x . ✓ Answer: 5 x (x + 2)

Example 3 — a negative common factor

Factorise - 6 x - 18 . 1 Both terms are negative and divide by 6, so use - 6 as the common factor. 2 Divide each term by - 6 : - 6 x \div (- 6) = x, and - 18 \div (- 6) = 3 . 3 Write the result: - 6 (x + 3) 4 Check by expanding: - 6 (x + 3) = - 6 x - 18 . ✓ Answer: - 6 (x + 3)

Example 4 — three terms

Factorise 8 x^{2} + 12 x - 4 . 1 Every term is a multiple of 4, so the highest common factor is 4 . 2 Divide each term by 4 : 8 x^{2} \div 4 = 2 x^{2}, 12 x \div 4 = 3 x, and - 4 \div 4 = - 1 . 3 Write the result: 4 (2 x^{2} + 3 x - 1) 4 Check by expanding: 4 (2 x^{2} + 3 x - 1) = 8 x^{2} + 12 x - 4 . ✓ Answer: 4 (2 x^{2} + 3 x - 1)

Practice

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What is the fully factorised form of 6 x + 15 ?

Factorise 9 x^{2} + 6 x fully, writing the answer as a(bx + c). Enter the value of a (the number outside the bracket).

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