One to one maths interventions built for GCSE success

Weekly online one to one GCSE maths revision lessons available in the spring term

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

Expanding brackets

Factors and multiples

Factors and multiples

Adding and subtracting Negative numbers

Multiplying Negative Numbers

Multiplying Algebraic Terms

Laws of Indices

Square Numbers

This topic is relevant for:

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.

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.

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)

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.

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.

**Example 1 – variable in just one term**

Fully factorise:

1Find the highest common factor (HCF) of the numbers 3 (the coefficient of

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

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

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

What do I need to multiply

What do I need to multiply

3(x + 2 )

We can check the answer by multiplying out the bracket!

**Example 2 – variable in just one term**

Fully factorise:

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

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

Fill in each term in the bracket by multiplying out.

What do I need to multiply **2**

What do I need to multiply **y**

We can check the answer by multiplying out the bracket!

**Example 3 – variable in two terms**

Fully factorise:

^{2} + 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 ^{2}

^{2} = ~~x~~ + x

~~x~~

The highest common factor (HCF) of ^{2}

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

Fill in each term in the bracket by multiplying out.

What do I need to multiply ^{2}

What do I need to multiply

We can check the answer by multiplying out the bracket!

^{2} + 12x

**Example 4 – variable in two terms**

Fully factorise:

^{2} – 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 ^{2}

^{2} = x ✕ ~~y~~

~~y~~

The highest common factor (HCF) of ^{2}**and**

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

Fill in each term in the bracket by multiplying out.

What do I need to multiply ^{2}

What do I need to multiply

We can check the answer by multiplying out the bracket!

^{2} – 10xy

**Example 5 – variable in two out of three terms**

Fully factorise:

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

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

Fill in each term in the bracket by multiplying out.

What do I need to multiply

What do I need to multiply

What do I need to multiply

We can check the answer by multiplying out the bracket!

**Example 6 – variable in two out of three terms**

Fully factorise:

^{2}y + 8xy^{2}

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 ^{2}^{2}y^{2}

~~x~~ ✕ ~~y~~

^{2}y = x ✕~~ x~~ ✕ ~~y~~

^{2} = ~~x~~ ✕ ~~y ~~✕ y

The highest common factor (HCF) of ^{2}^{2}y^{2}

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

Fill in each term in the bracket by multiplying out.

What do I need to multiply

What do I need to multiply ^{2}y

What do I need to multiply ^{2}

We can check the answer by multiplying out the bracket!

^{2}y + 8xy^{2}

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

- We must
**fully**factorise12x ^{2}– 6x = 2(6x^{2}– 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:12x ^{2}– 6x = 6x(2x – 1) 12x ^{2}– 6x

Even though this a quadratic expression we still factorise it to give a single bracket because it is not in the formax ^{2}+ bx + c- The term factorising can sometimes be written as ‘factoring’ or factorization’

1. Fully factorise: 5x + 10

Show answer

= 5(x + 2)

2. Fully factorise: 8 – 2y

Show answer

= 2(4 – y)

3. Fully factorise: 18x^{2} – 12x

Show answer

= 6x(3x – 2)

4. Fully factorise: 20y^{2} + 16xy

Show answer

= 4y(5y – 4x)

5. Fully factorise: 18 – 6y + 15x

Show answer

= 3(6 – 2y + 5x)

6. Fully factorise: 12y – 9x^{2} + 6y^{2}

Show answer

3y(4 – 3x^{2} + 2y)

(1)

1. Factorise 5x-20

Show answer

5(x-4)

(1 mark)

2. Factorise fully: 8x^{2} + 12xy

Show answer

4x(2x+3y)

(2 marks)

3. Factorise x^{2} + 8x

Show answer

x(x+8)

(1 mark)

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

- Manipulate algebraic expressions by taking out common factors to factorise into a single bracket.
~~Factorise quadratic expressions of the form x~~^{2}+ bx + c~~Factorise quadratic expressions of the form of the difference of two squares.~~~~Factorising quadratic expressions of the form ax~~^{2}+ bx + c (H)

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.