Back to course
1.3PureFoundation

Factorising Expressions

Factorising is expanding in reverse: you rewrite an expression as a product of brackets. It is the key that unlocks solving quadratics, simplifying fractions and sketching curves, so it is worth getting fast and accurate here.

35 min Algebraic Expressions
Edexcel AS Level Maths: 1.3 FactorisingCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Take out the highest common factor of an expression
  • Factorise quadratics of the form x² + bx + c
  • Factorise quadratics where the coefficient of x² is greater than 1
  • Recognise and factorise the difference of two squares
  • Choose the right factorising method for a given expression
1

Taking out a common factor

Always check for a common factor first. Look for the largest number and the highest power of each variable that divides into every term, then place it outside a bracket.

You can check your answer instantly by expanding — it should return the original expression.

ab+ac=a(b+c)ab + ac = a(b + c)
The shared factor aa comes out the front.
Example 1Factorise 6x2+9x6x^{2} + 9x
1The common factor of 66 and 99 is 33; both terms contain xx.
2So the factor is 3x3x.
3Divide each term by 3x3x: 6x2÷3x=2x6x^{2} \div 3x = 2x and 9x÷3x=39x \div 3x = 3.
Answer
3x(2x+3)3x(2x + 3)

Tip — Take out the HIGHEST common factor in one go. Pulling out just part of it (e.g. 3 instead of 3x) means more work and lost marks.

2

Factorising x² + bx + c

When the coefficient of x2x^{2} is 11, you need two numbers that multiply\textbf{multiply} to give cc (the constant) and add\textbf{add} to give bb (the coefficient of xx).

Pay attention to signs: a positive cc means both numbers share a sign; a negative cc means they have opposite signs.

x2+bx+c=(x+p)(x+q),  where p+q=b and pq=cx^{2} + bx + c = (x + p)(x + q), \;\text{where } p + q = b \text{ and } pq = c
Find two numbers that add to bb and multiply to cc.
Example 1Factorise x2+7x+12x^{2} + 7x + 12
1Need two numbers multiplying to 1212 and adding to 77.
2Those are 33 and 44.
Answer
(x+3)(x+4)(x + 3)(x + 4)
Example 2Factorise x2−2x−15x^{2} - 2x - 15
1Need two numbers multiplying to −15-15 and adding to −2-2.
2Those are −5-5 and +3+3.
Answer
(x−5)(x+3)(x - 5)(x + 3)

Tip — List the factor pairs of c, then see which pair adds to b. It turns guesswork into a quick checklist.

3

When the x² coefficient is bigger than 1

For ax2+bx+cax^{2} + bx + c with a>1a > 1, use the split-the-middle\textbf{split-the-middle} method: find two numbers that multiply to a×ca \times c and add to bb, split the middle term, then factorise in pairs.

Example 1Factorise 2x2+7x+32x^{2} + 7x + 3
1Multiply a×c=2×3=6a \times c = 2 \times 3 = 6. Need two numbers multiplying to 66, adding to 77: that is 66 and 11.
2Split the middle term: 2x2+6x+x+32x^{2} + 6x + x + 3.
3Factorise in pairs: 2x(x+3)+1(x+3)2x(x + 3) + 1(x + 3).
4The bracket (x+3)(x + 3) is common, so factor it out.
Answer
(2x+1)(x+3)(2x + 1)(x + 3)

Tip — If your two pairs do not share an identical bracket, recheck the numbers — one sign is usually wrong.

4

Difference of two squares

Any expression of the form “square minus square” factorises into a conjugate pair. Spotting this pattern is much faster than the general method.

a2−b2=(a+b)(a−b)a^{2} - b^{2} = (a + b)(a - b)
Two squares subtracted ⟶ a pair of brackets with opposite signs.
Example 1Factorise x2−25x^{2} - 25
1Recognise 25=5225 = 5^{2}, so this is x2−52x^{2} - 5^{2}.
Answer
(x+5)(x−5)(x + 5)(x - 5)
Example 2Factorise 9x2−169x^{2} - 16
19x2=(3x)29x^{2} = (3x)^{2} and 16=4216 = 4^{2}, so this is (3x)2−42(3x)^{2} - 4^{2}.
Answer
(3x+4)(3x−4)(3x + 4)(3x - 4)

Tip — Difference of two squares only works for subtraction. a² + b² does not factorise with real numbers.

Formula recap

ab+ac=a(b+c)ab + ac = a(b + c)
Common factor first — always.
x2+bx+c=(x+p)(x+q)x^{2} + bx + c = (x + p)(x + q)
p + q = b and pq = c.
a2−b2=(a+b)(a−b)a^{2} - b^{2} = (a + b)(a - b)
Difference of two squares.
2x2+7x+3=(2x+1)(x+3)2x^{2} + 7x + 3 = (2x + 1)(x + 3)
Harder quadratics: split the middle term.

Common mistakes to avoid

Taking out only part of the common factor: 6x² + 9x = 3(2x² + 3x).
Take out the full HCF 3x: 6x² + 9x = 3x(2x + 3).
Getting the signs wrong, e.g. x² − 2x − 15 = (x − 3)(x + 5).
The numbers must add to −2 and multiply to −15: (x − 5)(x + 3).
Trying to factorise a sum of two squares such as x² + 9.
x² + 9 does not factorise over the real numbers — leave it.

Key takeaways

  • Check for a common factor before anything else.
  • For x² + bx + c: two numbers that multiply to c and add to b.
  • For ax² + bx + c (a > 1): split the middle term and factorise in pairs.
  • Difference of two squares a² − b² = (a + b)(a − b) — subtraction only.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Factorise 8x + 12

    • 4(2x + 3)
    • 2(4x + 6)
    • 4(2x + 12)
    • 8(x + 12)
    Show answer and explanation

    Answer: 4(2x + 3)

    The highest common factor of 8 and 12 is 4.

  2. Factorise 10x² − 15x

    • 5x(2x − 3)
    • 5(2x² − 3x)
    • 5x(2x − 15)
    • x(10x − 15)
    Show answer and explanation

    Answer: 5x(2x − 3)

    HCF is 5x: 10x² ÷ 5x = 2x and 15x ÷ 5x = 3.

  3. Factorise x² + 9x + 20

    • (x + 4)(x + 5)
    • (x + 2)(x + 10)
    • (x + 1)(x + 20)
    • (x + 4)(x + 6)
    Show answer and explanation

    Answer: (x + 4)(x + 5)

    4 and 5 multiply to 20 and add to 9.

  4. Factorise x² − x − 12

    • (x − 4)(x + 3)
    • (x + 4)(x − 3)
    • (x − 6)(x + 2)
    • (x − 4)(x − 3)
    Show answer and explanation

    Answer: (x − 4)(x + 3)

    −4 and +3 multiply to −12 and add to −1.

  5. Factorise x² − 36

    • (x + 6)(x − 6)
    • (x − 6)²
    • (x + 36)(x − 1)
    • (x + 18)(x − 2)
    Show answer and explanation

    Answer: (x + 6)(x − 6)

    Difference of two squares: 36 = 6².

  6. Factorise 2x² + 7x + 3

    • (2x + 1)(x + 3)
    • (2x + 3)(x + 1)
    • (2x + 3)(x + 3)
    • (x + 1)(x + 3)
    Show answer and explanation

    Answer: (2x + 1)(x + 3)

    Split 7x into 6x + x: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

  7. Factorise 9x² − 25

    • (3x + 5)(3x − 5)
    • (9x + 5)(x − 5)
    • (3x − 5)²
    • (3x + 25)(3x − 1)
    Show answer and explanation

    Answer: (3x + 5)(3x − 5)

    (3x)² − 5², a difference of two squares.

  8. Factorise 3x² − 10x − 8

    • (3x + 2)(x − 4)
    • (3x − 2)(x + 4)
    • (3x + 4)(x − 2)
    • (3x − 4)(x + 2)
    Show answer and explanation

    Answer: (3x + 2)(x − 4)

    a × c = −24; −12 and +2 split the middle: 3x(x − 4) + 2(x − 4).

  9. Factorise fully 2x² − 18

    • 2(x + 3)(x − 3)
    • (2x + 6)(x − 3)
    • 2(x² − 9)
    • (x + 3)(2x − 6)
    Show answer and explanation

    Answer: 2(x + 3)(x − 3)

    Take out 2 first: 2(x² − 9), then difference of two squares.

  10. The area of a rectangle is x² + 5x + 6. Possible dimensions are:

    • (x + 2) and (x + 3)
    • (x + 1) and (x + 6)
    • (x + 5) and (x + 6)
    • (x + 2) and (x + 4)
    Show answer and explanation

    Answer: (x + 2) and (x + 3)

    x² + 5x + 6 = (x + 2)(x + 3), so the sides are (x + 2) and (x + 3).

Next step

Check your understanding

Work through questions on Factorising Expressions. Use the feedback to decide what needs another look.

1 past-paper question · 2 marks

Practise this topic

Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.