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Factorising to a Single Bracket

Here we will learn about factorising expressions into a single bracket.

There are also factorising worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is factorising?

Factorising is the reverse process of expanding brackets. To factorise an algebraic expression means to put it in brackets by taking out the common factors.

What is factorising?

Factorising

The reverse process of expanding brackets.

To factorise an algebraic expression means to put it in brackets by taking out the common factors.

E.g. Factorising

3x + 6 = 3(x + 2)

Welcome to Third Space Learning

How to factorise (single brackets)

In order to factorise an algebraic expression into a single bracket:

  1. Find the highest common factor of each of the terms in the expression.
  2. Write the highest common factor (HCF) at the front of a single bracket
  3. Fill in each term in the bracket by multiplying out.

Explain how to factorise (single brackets) in 3 steps

Factorising (single brackets)

In order to factorise an algebraic expression into a single bracket:

  1. Find the highest common factor of each of the terms in the expression.
  2. Write the highest common factor (HCF) at the front of a single bracket
  3. Fill in each term in the bracket by multiplying out.

Factorising examples (single brackets)

Example 1 – variable in just one term

Fully factorise:

3x + 6

1Find the highest common factor (HCF) of the numbers 3 (the coefficient of x) and 6 (the constant).

Factors of 3:
1, 3

Factors of 6
1, 6
2, 3

Top tip:
Writing the factor pairs makes it easier to list all the factors

The highest common factor (HCF) of 3x and 6 is 3

2Write the highest common factor (HCF) at the front of the single bracket.

3( + )

3Fill in each term in the bracket by multiplying out.

What do I need to multiply 3 by to give me 3x? x

What do I need to multiply 3 by to give me 6? 2

3(x + 2)

We can check the answer by multiplying out the bracket!

3(x + 2) = 3x + 6

Example 2 – variable in just one term

Fully factorise:

14 – 7y

Factors of 14:
1, 14
2, 7

Factors of 7:
1, 7

The highest common factor (HCF) of 14 and 7y is 7

7( + )

What do I need to multiply 7 by to give me 14? 2

What do I need to multiply 7 by to give me 7y? y

7(2 y) (don’t forget to keep the – here)

We can check the answer by multiplying out the bracket!

7(2y) = 14 – 7

Example 3 – variable in two terms

Fully factorise:

8x2 + 12x

Factors of 8:
1, 8
2,4

Factors of 12:
1, 12
2, 6
3, 4

Find the highest common factor (HCF) of the variables x2 and x

x2 = x + x

x = x

The highest common factor (HCF) of 8x2 and is 4x

4x( + )

What do I need to multiply 4x by to give me 8x2 ? 2x

What do I need to multiply 4x by give me 12x? 3

4x(2x + 3)

We can check the answer by multiplying out the bracket!

4x(2x + 3) = 8x2 + 12x

Example 4 – variable in two terms

Fully factorise:

15y2 – 10xy

Factors of 15:
1, 15
3, 5

Factors of 10:
1, 10
2, 5

Find the highest common factor (HCF) of the letters 15y2 and xy

y2 = x ✕ y

xy = x ✕ y

The highest common factor (HCF) of 15y2 and 10xy is 5y

5y ( – )

What do I need to multiply 5y by to give me 15y2? 3y

What do I need to multiply 5y by to give me 10xy? 2x

5y(3y2x)

We can check the answer by multiplying out the bracket!

5y(3y2x) = 15y2 – 10xy

Example 5 – variable in two out of three terms

Fully factorise:

6x + 2y – 12

Factors of 6:
1, 6
2, 3

Factors of 2:
1, 2

Factors of 12:
1, 12
2, 6
3,4

The highest common factor (HCF) of 6 and 2 and 12 is 2

2 ( + – )

What do I need to multiply 2 by to give me 6x? 3x

What do I need to multiply 2 by to give me 2y? y

What do I need to multiply 2 by to give me 12? 6

2(3x + y6) Don’t forget to keep the – here!

We can check the answer by multiplying out the bracket!

2(3x+ y6) = 6x + 2y – 12

Example 6 – variable in two out of three terms

Fully factorise:

12xy – 4x2y + 8xy2

Factors of 12:
1, 12
2, 6
3,4

Factors of 4:
1, 4
2, 2

Factors of 8:
1, 8
2, 4

Find the highest common factor (HCF) of the letters x2 and x2y and xy2

xy = xy

x2y = x ✕ xy

xy2 = xy ✕ y

The highest common factor (HCF) of 12x2 and 4x2y and 8xy2 is 4xy

4xy( – + )

What do I need to multiply 4xy by to give me 12xy? 3

What do I need to multiply 4xy by to give me 4x2y? x

What do I need to multiply 4xy by to give me 8xy2? 2y

4xy(3x + 2y) Don’t forget to keep the – here!

We can check the answer by multiplying out the bracket!

4xy(3 x + 2y) = 12xy – 4x2y + 8xy2

Common Misconceptions

These are some of the common misconceptions around factorising into single brackets

  • We must fully factorise

    12x2 – 6x = 2(6x2 – 3x)

    Here we have factorised the expression, however it is not fully factorised because 2 is not the highest common factor.

    6x is the highest common factor, so this is the correct final answer:

    12x2 – 6x = 6x(2x – 1)

  • 12x2 – 6x

    Even though this a quadratic expression we still factorise it to give a single bracket because it is not in the form ax2 + bx + c

  • The term factorising can sometimes be written as ‘factoring’ or factorization’

Practice Factorising Questions (Single Brackets)

1. Fully factorise: 5x + 10

Show answer

= 5(x + 2)

2. Fully factorise: 8 – 2y

Show answer

= 2(4 – y)

3. Fully factorise: 18x2 – 12x

Show answer

= 6x(3x – 2)

4. Fully factorise: 20y2 + 16xy

Show answer

= 4y(5y – 4x)

5. Fully factorise: 18 – 6y + 15x

Show answer

= 3(6 – 2y + 5x)

6. Fully factorise: 12y – 9x2 + 6y2

Show answer

3y(4 – 3x2 + 2y)

(1)

GCSE Factorising Questions (Single Brackets)

1. Factorise 5x-20

Show answer

5(x-4)

(1 mark)

2. Factorise fully: 8x2 + 12xy

Show answer

4x(2x+3y)

(2 marks)

3. Factorise x2 + 8x

Show answer

x(x+8)

(1 mark)

Factorising worksheets

Download for free one or more factorising worksheets to help you practise this further and more

Learning Checklist:

  • Manipulate algebraic expressions by taking out common factors to factorise into a single bracket.
  • Factorise quadratic expressions of the form x2 + bx + c
  • Factorise quadratic expressions of the form of the difference of two squares.
  • Factorising quadratic expressions of the form ax2 + bx + c (H)

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