Expanding Brackets

Here we break down everything you need to know about expanding brackets. You’ll learn how to expand single brackets and double brackets in order to leave a simplified algebraic expression.

At the end you’ll find expanding brackets worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

This lesson is part of our series on algebraic expressions. You may find it helpful to start with the main algebraic expressions lesson for a summary of what to expect and then also work through the following:

What does expanding brackets mean?

Expanding brackets is the process by which we remove brackets.

Expanding brackets is the reverse process of factorisation and is sometimes referred to as multiplying out.

To expand brackets we multiply everything outside of the bracket, by everything inside of the bracket.

There are three main types of expanding brackets, each of which is covered below:

Expanding single brackets

3(2x + 1) = 6x + 3

Expressions with two terms like 6x + 3 are known as a binomials.

Expanding double brackets

(x + 5)(x – 1) = x2+ 4x – 5

Expressions with three terms like x2 + 4x − 5 are known as trinomials.

Terms that are raised to the power of 2 like x2 are known as quadratic terms.

Expanding triple brackets

(x + 1)(x + 2)(x + 3) = x3 + 6x + 11x + 6

A polynomial expression consists of two or more algebraic terms.

1) Expanding single brackets

To expand a single bracket we multiply the term outside of the bracket by everything inside of the bracket.

How to expand single brackets

In order to expand single brackets:

1. Multiply the term outside of the bracket by the first term inside the bracket.
2. Multiply the term outside the bracket by the second term inside the bracket.

Expanding brackets examples (single brackets)

Example 1: two terms in the bracket

Expand:

2(x + 3)

1. Multiply the term outside of the bracket (2) by the first term inside the bracket (x).

2 ✕ x = 2x

2Multiply the value outside the bracket (2) by the second term inside the bracket (3).

2 ✕ 3 = 6

The answer is positive so we need to write + 6.

2(x + 3) = 2x + 6

Example 2: two terms in the bracket and a negative term outside

Expand:

− 3(y − 4)

− 3 ✕ y = − 3y

+ = so the answer is negative.

− 3 x − 4 = + 12

= + so the answer is positive. We need to write + 12.

− 3(y − 4) = − 3y + 12

Example 3: two terms in the bracket and variables with coefficients outside

Expand:

3x(4x − 2y)

3x ✕ 4x = 12x2

3x ✕ − 2y = − 6xy

+ = so the answer is negative. We need to write − 6xy.

3x(4x − 2y) = 12x2 − 6xy

Example 4: three terms in the bracket and variables with coefficients greater than 1

Expand:

2x(3 − 5y + 6x2)

2x ✕ 3 = 6x

2x ✕ − 5y = − 10xy

+ = so the answer is negative. We need to write 10xy.

2x ✕ 6x2 = + 12x3

The answer is positive so we need to write +12x3.

$2x( 3 – 5y + 6x^2) =$
$6x - 10xy + 12x^3$

Practice expanding brackets questions (single brackets)

1. Expand: 6(y − 2)

= 6y − 12

2. Expand: − 2(x + 6)

= - 2x − 12

3. Expand: y(2y − 3)

= 2y2 − 3y

4. Expand: - 5x(2x + 4)

= − 10x2 − 20x

5. Expand: 4(8 − 3x + 2y)

= 32 − 12x + 8y

6. Expand: 3y(2 + 7y − 4x)

= 6y + 21y2 − 12xy

2) Expanding double brackets

To expand double brackets we multiply every term in the first bracket, by every term in the second bracket.

How to expand double brackets

In order to expand double brackets follow these steps:

1. Draw a grid and insert the terms of the first and second brackets.
2. Fill in the grid by multiplying each of the terms together.
3. Write out each of the terms and simplify the expression by collecting like terms.

Explain how to expand double brackets in 3 steps

Expanding double brackets

In order to expand double brackets:

1. Draw a grid and insert the terms of the first and second brackets.
2. Fill in the grid by multiplying each of the terms together.
3. Write out each of the terms and simplify the expression by collecting like terms.

Expanding brackets examples (double brackets)

Example 1: variables have a coefficient of 1 and there is + in both brackets

Expand and simplify:

(x+2)(x+3)

1. Draw a grid and insert the terms of the first and second brackets.

2Fill in the grid my multiplying each of the terms together.

x ✕ x = x2
x ✕ 3 = 3x
x ✕ 2 = 2x
2 ✕ 3 = 6

3Write out each of the terms and simplify the expression by collecting like terms.

x2 + 3x + 2x + 6
x2 + 5x + 6

Example 2: variables have a coefficient of 1 and there is + in one bracket and - in the other

Expand and simplify:

(x + 5)(x − 1)

x ✕ x = x2
x ✕ − 1 = − x

+ = so the answer is negative.

x ✕ 5 = 5x
5 ✕ − 1 = − 5

+ = so the answer is negative.

x2 − x + 5x − 5
x2 + 4x − 5

Example 3: variables have a coefficient greater than 1 and there is + in one bracket and - in the other

Expand and simplify:

(2x − 3)(x + 4)

2x ✕ x = 2x2
2x ✕ 4 = 8x
x ✕ − 3 = − 3x

+ = so the answer is negative.

4 ✕ − 3 = − 12

+ = so the answer is negative.

2x2 + 8x − 3x − 12
2x2 + 5x − 12

Example 4: squared brackets

Expand and simplify:

(3x − 4)2

(3x − 4)2 = (3x − 4)(3x − 4)

Remember: when we square something (raise it to the power of 2) we multiply it by itself.

3x ✕ 3x = 9x2
3x ✕ − 4 = − 12x
3x ✕ − 4 = − 12x

+ = so the answer is negative.

− 4 ✕ − 4 = + 16

= + so the answer is positive.

9x2 − 12x − 12x + 16
9x2 − 24x + 16

Practice expanding brackets questions (double brackets)

1. Expand and simplify: (x + 5)(x + 6)

= x2 + 11x + 30

2. Expand and simplify: (x − 4)(x + 2)

= x2 − 2x − 8

3. Expand and simplify: (2x + 3)(x + 4)

=2x2 + 11x + 12

4. Expand and simplify: (3x − 2)(x + 1)

= 3x2 + x − 2

5. Expand and simplify: (x − 4)2

= x2 − 8x + 16

6. Expand and simplify: (2x + 5)2

= 4x2 + 20x + 25

3) Expanding triple brackets

To expand triple brackets we first multiply the first two brackets together. We then multiply every term in this new expression by every term in the third bracket.

How to expand triple brackets

In order to expand triple brackets:

1. Draw a grid, insert the terms of the first and second brackets, then fill it in by multiplying each of the terms together.
2. Write out each of the terms and simplify the expression by collecting like terms.
3. Draw a grid, insert the terms from this new expression and the third bracket, then fill it in by multiplying each of the terms together.
4. Write out each of the terms and simplify the expression by collecting like terms.

Expanding brackets examples (triple brackets)

Example 1: multiply three brackets

Expand and simplify:

(x + 1)(x + 2)(x + 3)

1. Draw a grid, insert the terms of the first and second brackets, then fill it in by multiplying each of the terms together.

x ✕ x = x2
x ✕ 2 = 2x
x ✕ 1 = x
1 ✕ 2 = 2

2Write out each of the terms and simplify the expression by collecting like terms.

x2 + 2x + x + 2
x2 + 3x + 2

3Draw a grid, insert the terms from this new expression and the third bracket, then fill it in by multiplying each of the terms together.

x ✕ x2 = x3
x ✕ 3x = 3x2
x ✕ 6 = 6x
3 ✕ x2 = 3x2
3 ✕ 3x = 9x
3 ✕ 6 = 18

4Write out each of the terms and simplify the expression by collecting like terms.

x3 + 3x2 + 3x2 + 9x + 2x + 6
x3 + 6x2 + 11x + 6

Example 2: squared brackets multiplied by a third bracket

Expand and simplify:

(x + 3)2(x − 1)

(x + 3)2 = (x + 3)(x + 3)

x ✕ x = x2
x ✕ 3 = 3x
x ✕ 3 = 3x
3 ✕ 3 = 9

x2 + 3x + 3x + 9
x2 + 6x + 9

x ✕ x2 = x3
x ✕ 6x = 6x2
x ✕ 9 = 9x
− 1 ✕ x2 = − x2
− 1 ✕ 6x = − 6x
− 1 ✕ 9= − 9

x3 + 6x2 − x2 + 9x − 6x − 9
x3 + 5x2 + 3x − 9

Practice expanding brackets questions (triple brackets)

1. Expand and simplify: (x + 2)(x + 3)(x + 4)

= x3 + 9x2 + 26x + 24

2. Expand and simplify: (x + 3)(x − 2)2

= x3 − x2 − 8x + 12

3. Expand and simplify: (2x + 1)3

= 8x3 + 12x2 + 6x + 1

Common misconceptions

• We must multiply the value outside the brackets by every term inside the brackets.
2(6x2 − 3x) = 12x2 − 3x  ✖

Here we have multiplied the value outside of the brackets by the first term inside of the bracket, but not the second term.

The correct answer is 2(6x2 − 3x) = 12x2 − 6x

We need to multiply all the terms inside the bracket.

• For two numbers to multiply to give a + their signs must be the same.

+ ✕ + = +
− ✕ − = +

e.g. 2 ✕ 3 = 6
e.g. − 2 ✕ − 3 = 6

4 ✕ 5 = 20
− 4 ✕ − 5 = 20

• For two numbers to multiply to give a their signs must be different.

+ ✕ − = −
− ✕ + = −

e.g. 2 ✕ − 3 = − 6
e.g. − 2 ✕ 3 = − 6

4 ✕ − 5 = − 20
− 4 ✕ 5 = − 20

− 4(3y − 5) = − 4y − 20 ✖

Here we have not used = +

− 4 ✕ − 5 = + 20

The correct answer is − 4(3y − 5) = − 4y + 20

• When we square something, we multiply it by itself.
32 = 3 ✕ 3
x2 = x ✕ x
(5y)2 = 5y ✕ 5y

• When we square a bracket, we multiply it by the entire bracket.
(x + 3)2 = (x + 3)(x + 3) ✔
NOT x2 + 9 ✖

Expanding brackets GCSE questions

1. Expand: 3(x - 2)

3x - 6

(1 mark)

2. Expand: 4x(2x - 7)

8x2 - 28x

(1 mark)

3. Expand and simplify: 5(x - 3) - 3(x + 5)

2x - 30

(2 marks)

Expanding brackets worksheet

Download a free expanding brackets worksheet with 20+ reasoning and applied questions, answers and mark scheme to help your students prepare for GCSEs. Includes reasoning and applied questions.

Learning checklist

You have now learned how to:

• Multiply a single term over a bracket
• Expand products of 2 or more binomials

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