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In order to access this I need to be confident with:

Expanding brackets

Factors and multiples

Adding and subtracting Negative numbers

Multiplying Negative Numbers

Multiplying Algebraic Terms

Laws of Indices

Square Numbers

This topic is relevant for:

GCSE Maths Algebra Factorising Single Bracket

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)

How to factorise (single brackets)

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

Find the highest common factor of each of the terms in the expression.
Write the highest common factor (HCF) at the front of a single bracket
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:

Find the highest common factor of each of the terms in the expression.
Write the highest common factor (HCF) at the front of a single bracket
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

Find the highest common factor (HCF) of the numbers 14 and 7.

Factors of 14:
1, 14
2, 7

Factors of 7:
1, 7

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

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

7( + )

Fill in each term in the bracket by multiplying out.

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(2 – y) = 14 – 7

Example 3 – variable in two terms

Fully factorise:

8x² + 12x

Find the highest common factor (HCF) of the numbers 8 and 12.

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

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

4x( + )

Fill in each term in the bracket by multiplying out.

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

Find the highest common factor (HCF) of the numbers 15 and 10.

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

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

5y ( – )

Fill in each term in the bracket by multiplying out.

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(3y – 2x)

We can check the answer by multiplying out the bracket!

5y(3y – 2x) = 15y² – 10xy

Example 5 – variable in two out of three terms

Fully factorise:

6x + 2y – 12

Find the highest common factor (HCF) of the numbers 6, 2 and 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

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

2 ( + – )

Fill in each term in the bracket by multiplying out.

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 + y – 6) Don’t forget to keep the – here!

We can check the answer by multiplying out the bracket!

2(3x+ y – 6) = 6x + 2y – 12

Example 6 – variable in two out of three terms

Fully factorise:

12xy – 4x²y + 8xy²

Find the highest common factor (HCF) of the numbers 12, 4 and 8

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 = x ✕ y

x²y = x ✕ x ✕ y

xy² = x ✕ y ✕ y

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

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

4xy( – + )

Fill in each term in the bracket by multiplying out.

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(3 – x + 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

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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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What is factorising?

How to factorise (single brackets)

Factorising examples (single brackets)

Common Misconceptions

Factorising Questions (Single Brackets)

Factorising GCSE Questions (Single Brackets)

Factorising Worksheets

Learning Checklist

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