Find a common numerical or literal factor

Common numerical factor

To factorise a sum, we start by identifying what appears identically in each term.

Example: factorise 10x + 15.

• 10x = 5 × 2x
• 15 = 5 × 3 The common factor is 5. We write: 10x + 15 = 5(2x + 3).

Common factor involving a letter

A letter can also be a common factor.

Example: factorise 5x + x².

• 5x = x × 5
• x² = x × x The common factor is x. We write: 5x + x² = x(5 + x).

3-step method

1. Identify what is repeated in each term (number and/or letter)
2. Write each term as a product containing this common factor
3. Highlight the common factor in front of a set of brackets

Complete example

Factorise 12x + 8:

• The common factor of 12 and 8 is 4
• 12x = 4 × 3x, 8 = 4 × 2
• 12x + 8 = 4(3x + 2)

Common pitfall

A common pitfall is to choose a common factor that is not the largest possible (factoring by 2 instead of 4, for example), which gives a correct result but one that is not fully factored. Always look for the largest possible common factor, and check by expanding the expression to ensure you have indeed recovered the original expression.