1MA1 · A4AlgebraFoundation & HigherGrades 3–7

Factorising Expressions

Q: Does the order of the brackets matter?

No. (x + 2)(x + 3) and (x + 3)(x + 2) are the same.

Factorising writes an expression as a product — the reverse of expanding. $x^{2} + 5 x + 6$ becomes $(x + 2) (x + 3)$ . It looks like tidying up, but it is one of the most useful tools in algebra: it solves quadratic equations, simplifies algebraic fractions and reveals where graphs cross the axis.

60 min Algebraic Expressions

The big picture

There are three patterns to recognise, and a fixed order to check them in: first take out any common factor, then look for a difference of two squares, then try to factorise as a quadratic. Checking in that order prevents incomplete answers, which lose marks even when the brackets are correct.

Every factorisation can be checked by expanding it again. That makes this a topic where you should never lose a mark to an unnoticed error.

What you'll learn

Factorise by taking out the highest common factor
Factorise quadratics of the form $x^{2} + b x + c$
Factorise the difference of two squares
Factorise quadratics of the form $a x^{2} + b x + c$ (Higher)

Common factors

Find the highest common factor of every term — both the number and the letters — and write it outside the bracket.

$12 x^{2} y - 18 x y^{2} = 6 x y (2 x - 3 y)$ .

A factorisation is only complete when the terms inside the bracket share no further common factor.

1HCF of 15, 25 and 5 is 5; lowest powers

a^{2} b

2Divide each term by

5 a^{2} b

3 a

5 b

- 1

Answer

5 a^{2} b (3 a + 5 b - 1)

Your turnSame method, new numbers — try it before moving on.

Factorise completely $12 x^{2} y - 18 x y^{2}$ .

Your answer

Tip — If a term is the whole common factor, it leaves a $1$ (or $- 1$ ) inside the bracket. Leaving it out is a common slip.

Quadratics with a = 1

To factorise $x^{2} + b x + c$ , find two numbers that $multiply to$ $c$ and $add to$ $b$ .

For $x^{2} - 2 x - 15$ : the pair $- 5$ and $3$ multiply to $- 15$ and add to $- 2$ , so it factorises as $(x - 5) (x + 3)$ .

Sign clues: if $c$ is positive, both numbers have the same sign as $b$ ; if $c$ is negative, the numbers have opposite signs.

1Need a product of

+ 24

and a sum of

- 11

: both negative.

- 3

and

- 8

work.

Answer

(x - 3) (x - 8)

Your turnSame method, new numbers — try it before moving on.

Factorise $x^{2} + 2 x - 35$ .

Your answer

Difference of two squares

Any expression of the form $a^{2} - b^{2}$ factorises as $(a + b) (a - b)$ .

Look for two perfect squares with a minus between them: $49 x^{2} - 16 = (7 x + 4) (7 x - 4)$ .

Take out common factors first: $3 x^{2} - 75 = 3 (x^{2} - 25) = 3 (x + 5) (x - 5)$ .

a^{2} - b^{2} = (a + b) (a - b)

Difference of two squares.

Your turnSame method, new numbers — try it before moving on.

Factorise $25 y^{2} - 49$ .

Your answer

Why this works

There is no $x$ term because the middle terms cancel when you expand $(a + b) (a - b)$ . That missing middle term is exactly how you spot this pattern. A sum of two squares, like $x^{2} + 9$ , does not factorise at all.

Quadratics with a ≠ 1

For $a x^{2} + b x + c$ , find two numbers that multiply to $a c$ and add to $b$ . Use them to split the middle term, then factorise in pairs.

For $3 x^{2} + 10 x + 8$ : $a c = 24$ , and $6$ and $4$ add to $10$ . So $3 x^{2} + 6 x + 4 x + 8 = 3 x (x + 2) + 4 (x + 2) = (3 x + 4) (x + 2)$ .

a c = - 30

; numbers

- 10

and

3

multiply to

- 30

and add to

- 7

2Split:

6 x^{2} - 10 x + 3 x - 5

3Pairs:

2 x (3 x - 5) + 1 (3 x - 5)

Answer

(2 x + 1) (3 x - 5)

Your turnSame method, new numbers — try it before moving on.

Factorise $3 x^{2} + 10 x - 8$ .

Your answer

Tip — After splitting and pairing, both pairs must contain the same bracket. If they do not, recheck the split or the signs.

Think like an examiner

"Factorise fully" means take out every common factor first.
Check by expanding your answer.
Show the split middle term for $a x^{2} + b x + c$ quadratics.

Factorising

x^{2} + b x + c = (x + p) (x + q)

Where

pq = c

and

p + q = b

a^{2} - b^{2} = (a + b) (a - b)

Difference of two squares.

a x^{2} + b x + c : split b using a c

Non-unit leading coefficient.

Watch out for these

Factorising

4 x^{2} - 8 x

4 (x^{2} - 2 x)

Not complete:

4 x (x - 2)

Writing

x^{2} - 9 = (x - 3)^{2}

Difference of two squares:

(x + 3) (x - 3)

Choosing numbers that add to

c

and multiply to

b

Multiply to

c

(or

a c

), add to

b

Trying to factorise

x^{2} + 16

A sum of squares does not factorise over the integers.

Stretch yourself

Show that $n^{3} - n$ is always divisible by 6 for any integer $n$ .

Hint — Factorise completely: take out $n$ , then use the difference of two squares. Think about consecutive integers.

Questions students ask

How do I know when an expression is fully factorised?

When no bracket can be factorised further and there is no common factor left inside any bracket.

Does the order of the brackets matter?

No.

(x + 2) (x + 3)

and

(x + 3) (x + 2)

are the same.

What if I cannot find two numbers that work?

The quadratic may not factorise. For equations, use the quadratic formula instead.

Key takeaways

Always take out the highest common factor first.
For $x^{2} + b x + c$ , find numbers multiplying to $c$ and adding to $b$ .
$a^{2} - b^{2} = (a + b) (a - b)$ .
For $a x^{2} + b x + c$ , split the middle term using $a c$ and factorise in pairs.
Check every factorisation by expanding.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 33 marks

(a)Factorise $8 x - 20$ .[1]
(b)Factorise fully $9 a^{2} b + 6 a b^{2}$ .[2]

Question 2Grade 53 marks

(a)Factorise $x^{2} - 3 x - 40$ .[2]
(b)Hence solve $x^{2} - 3 x - 40 = 0$ .[1]

Question 3Grade 72 marks

Factorise fully $2 x^{3} - 50 x$ .[2]

Unlimited practice

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Factorising Expressions

Common factors

Quadratics with a = 1

Difference of two squares

Quadratics with a ≠ 1

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Related topics

Factorising Expressions

Common factors

Quadratics with a = 1

Difference of two squares

Quadratics with a ≠ 1

Practise it the way it is examined

Skill drill

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