Factorising is the reverse of expanding: it writes an expression as a product. To take out a common factor, find the highest common factor of every term and put it outside a bracket, such as 6x + 9 = 3(2x + 3). To factorise x² + bx + c, find two numbers that multiply to c and add to b. Expressions of the form x² − a² factorise as (x − a)(x + a).
Factorising methods
- Common factor: take out the HCF of every term
- x² + bx + c = (x + p)(x + q), where p × q = c and p + q = b
- Difference of two squares: x² − a² = (x − a)(x + a)
- Check by expanding your answer
Taking out a common factor
Find the highest common factor of all the terms and write it outside a bracket; inside the bracket goes what is left of each term. For 6x + 9 the highest common factor is 3, so 6x + 9 = 3(2x + 3). Expand your answer to check it.
Factorising fully, with letters
A common factor can include letters. Every term of x² + 5x contains x, so x² + 5x = x(x + 5). “Factorise fully” means take out the highest common factor of both numbers and letters: 12x² − 8x = 4x(3x − 2), not 2x(6x − 4) or 4(3x² − 2x).
Factorising quadratics
To factorise x² + bx + c, find two numbers that multiply to give c and add to give b. For x² + 9x + 20, the numbers 4 and 5 multiply to 20 and add to 9, so x² + 9x + 20 = (x + 4)(x + 5). If c is negative, one number is positive and one negative: x² − 2x − 15 = (x − 5)(x + 3).
Difference of two squares
An expression made of one square minus another factorises as x² − a² = (x − a)(x + a). For example, x² − 49 = (x − 7)(x + 7). There is no x term because the middle terms cancel when you expand.
Quadratics with a coefficient of x² (Higher)
For ax² + bx + c with a ≠ 1, find two numbers that multiply to a × c and add to b, then split the middle term. For 2x² + 7x + 3, a × c = 6, and 6 and 1 add to 7: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Worked examples
Example 1. Factorise 6x + 9.
- The highest common factor is 3.
- 6x ÷ 3 = 2x and 9 ÷ 3 = 3.
Answer: 3(2x + 3).
Example 2. Factorise fully 12x² − 8x.
- The highest common factor is 4x.
- 12x² ÷ 4x = 3x and 8x ÷ 4x = 2.
Answer: 4x(3x − 2).
Example 3. Factorise x² + 9x + 20.
- Two numbers that multiply to 20 and add to 9: 4 and 5.
Answer: (x + 4)(x + 5).
Example 4. Factorise x² − 2x − 15.
- Multiply to −15 and add to −2: −5 and 3.
Answer: (x − 5)(x + 3).
Example 5. Factorise x² − 49.
- This is a difference of two squares: x² − 7².
Answer: (x − 7)(x + 7).
Example 6. Factorise 2x² + 7x + 3.
- a × c = 6; numbers that multiply to 6 and add to 7: 6 and 1.
- 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3).
Answer: (2x + 1)(x + 3).
Common mistakes
- Not factorising fully. Take out the highest common factor, including any letters.
- Choosing numbers that add to c. The pair must multiply to c and add to b.
- Sign errors with a negative constant. If c is negative, one number is negative.
- Missing a difference of two squares. x² − 49 factorises even though there is no x term.
- Not checking by expanding. Expand your brackets to confirm the original expression.
How factorising is examined
Foundation papers in 1MA1 and 4MA1 test common factors and simple quadratics, while Higher papers add ax² + bx + c. Questions ask you to:
- factorise using a common factor, sometimes “fully”
- factorise a quadratic of the form x² + bx + c
- factorise a difference of two squares
- factorise a quadratic with a coefficient of x² other than 1 (Higher)
The factorising worksheet includes answers, and the GCSE algebra revision engine gives more practice.
Factorising reverses expanding brackets, and it is used to solve quadratic equations and simplify algebraic fractions.
Frequently asked questions
What does it mean to factorise?
To write an expression as a product, usually by putting a common factor outside a bracket or by writing a quadratic as two brackets.
How do you factorise x² + bx + c?
Find two numbers that multiply to c and add to b; they go in the two brackets.
What is the difference of two squares?
x² − a² = (x − a)(x + a).