Expanding and Factorising

What you'll learn

How brackets such as $3 (x + 2)$ show multiplication.
How to expand: remove brackets by multiplying.
How to factorise: put brackets back in by taking out something all parts share.
How to simplify an expression and solve an equation with brackets.

The algebra basics

Before you start expanding and factorising, make sure the algebra words feel familiar.

Definition

Key algebra words

An expression is algebra with no equals sign, such as $4 x + 7$ .
A variable is a letter that stands for a number, such as $x$ or $y$ .
A term is one part of an expression, separated by plus or minus signs. Include the sign in front.
A coefficient is the number multiplying a variable. In $6 a$ , the coefficient is 6.
A constant is a term with no variable, such as 9.
A factor is something being multiplied. In $5 (t + 2)$ , 5 and $t + 2$ are factors.
A power tells you repeated multiplication: $x^2$ means $\times x$ .

Like terms have the same variable part, such as $3 x$ and $8 x$ . You can collect like terms by adding or subtracting their coefficients.

Example

Collecting like terms

Start with $8 x + 5 x - 4$ .
The like terms are $8 x$ and $5 x$ because both have $x$ .
Add the coefficients:
$8 x + 5 x = 13 x$
The constant stays separate, so the simplified expression is $13 x - 4$ .

Expanding one bracket

Definition

Expand

To expand means to remove brackets by multiplying the factor outside the bracket by every term inside the bracket.

This rule is called the distributive law: the outside factor is distributed to each term inside the bracket.

Key Idea

Brackets mean multiply

In $a (b + c)$ , the outside factor $a$ multiplies both $b$ and $c$ , not just the first term.

The distributive law shows the outside factor multiplying every term inside the bracket.

Example

Expanding a simple bracket

Start with $6 (3 x + 4)$ . The outside factor is 6.

The 6 is distributed to both terms inside the bracket.

Multiply 6 by each term inside the bracket.
$\times 3x + 6 \times 4$
Work out each multiplication.
$6 (3 x + 4) = 18 x + 24$

Common Mistake

Only multiplying the first term

Do not leave the second term unchanged: $6 (3 x + 4)$ is not $18 x + 4$ . The 6 multiplies the 4 as well.

A common error is multiplying only the first term and forgetting the second term.

Expanding when there is a letter outside

The outside factor might include a letter, such as $3 p$ . Multiply it by each term in the bracket.

Example

Expanding with a letter outside

Start with $3 p (p - 5)$ . The outside factor is $3 p$ .

When the outside factor has a letter, it still multiplies every term inside the bracket.

Multiply $3 p$ by $p$ .
$3p \times p = 3p^2$
Multiply $3 p$ by $- 5$ .
$3p \times (-5) = -15p$
Put the two results together: $3p^2 - 15p$ .

Factorising: putting brackets back in

Definition

Factorise

To factorise means to write an expression using brackets by taking out a common factor. It is the reverse of expanding.

A common factor is a number, letter, or expression that is a factor of every term. The highest common factor, or HCF, is the largest common factor you can take out.

Example

Factorising a simple expression

Start with $8 r + 20$ .

Factorising takes the common factor out and places the remaining parts inside a bracket.

The HCF of 8 and 20 is 4.
Rewrite each term using 4.
$8 r + 20 = 4 (2 r) + 4 (5)$
Put the common factor outside the bracket.
$8 r + 20 = 4 (2 r + 5)$

Tip

Check by expanding

To check factorising, expand your answer. If $4 (2 r + 5)$ gives $8 r + 20$ , your brackets are correct.

Factorising fully with letters

To factorise fully means take out the biggest common factor, including any letters that appear in every term.

Example

Factorising fully

Start with $14a^2b + 21ab^2$ .

Factorising fully means taking out the full common factor shared by both terms.

The HCF of 14 and 21 is 7.
Both terms contain $a$ and $b$ , so the letter part of the common factor is $ab$ .
The full common factor is $7 ab$ . Divide each term by $7 ab$ .
$14a2b÷7ab=2a21ab2÷7ab=3b\begin{aligned} 14a^2b \div 7ab &= 2a\\ 21ab^2 \div 7ab &= 3b \end{aligned}$
Write the answer as $7 ab (2 a + 3 b)$ .

Common Mistake

Stopping too early

$14a^2b+21ab^2=7(2a^2b+3ab^2)$ is not fully factorised because $a$ and $b$ can still be taken out.

Expanding and simplifying

To simplify means to write an expression in a shorter form, usually by collecting like terms. For these questions, expand first, then collect like terms.

Example

Expand and simplify

Start with $5 (s + 3) + 2 (s - 4)$ .

Expanding both brackets first makes the like terms easy to collect.

Expand both brackets.
$5 (s + 3) + 2 (s - 4) = 5 s + 15 + 2 s - 8$
Collect the like terms.
$5 s + 2 s + 15 - 8 = 7 s + 7$

Common Mistake

Minus before a bracket

A minus sign before a bracket affects every term inside it. For example, $- 3 (y + 5)$ becomes $- 3 y - 15$ , not $- 3 y + 15$ .

Example

Subtracting a bracket

Start with $4 (2 y - 3) - 3 (y + 5)$ .

The minus before the second bracket means the outside factor is −3, so it multiplies both terms inside.

Expand both brackets. The second bracket has outside factor $- 3$ .
$4 (2 y - 3) - 3 (y + 5) = 8 y - 12 - 3 y - 15$
Collect like terms.
$8 y - 3 y - 12 - 15 = 5 y - 27$

Solving equations with brackets

An equation has an equals sign. To solve an equation means to find the value of the variable that makes both sides equal.

An inverse operation is an undo step. For example, division undoes multiplication, and subtraction undoes addition.

Tip

Divide first

If the whole bracket is multiplied by a number, dividing first is usually quicker than expanding.

Example

Solving a bracket equation

Start with $6 (d + 2) = 48$ . The bracket is multiplied by 6, so divide both sides by 6.
$d + 2 = 8$
Undo the plus 2 by subtracting 2 from both sides.
$d = 6$
Check in the original equation: $6 (6 + 2) = 48$ .

Example

When the answer is not a whole number

Start with $2 (v - 6) = 15$ . Divide both sides by 2.
$v - 6 = 7.5$
Undo the minus 6 by adding 6 to both sides.
$v = 13.5$
A decimal answer is fine if the division does not come out as a whole number.

Exam technique

In the exam

When expanding, make sure every term inside the bracket has been multiplied by the outside factor.
When factorising, look for the highest common factor and check by expanding your answer.
When solving equations, do the same inverse operation to both sides and check your value in the original equation.

Self review

Check yourself

Can you expand $5 (2 x - 3)$ without missing the negative sign?
Can you fully factorise $12a^2 + 18ab$ by taking out a number and a letter?
If $3 (k - 4) = 21$ , what is the first operation you would undo?

Expanding and Factorising

What you'll learn

How brackets such as $3 (x + 2)$ show multiplication.
How to expand: remove brackets by multiplying.
How to factorise: put brackets back in by taking out something all parts share.
How to simplify an expression and solve an equation with brackets.

The algebra basics

Before you start expanding and factorising, make sure the algebra words feel familiar.

Definition

Key algebra words

An expression is algebra with no equals sign, such as $4 x + 7$ .
A variable is a letter that stands for a number, such as $x$ or $y$ .
A term is one part of an expression, separated by plus or minus signs. Include the sign in front.
A coefficient is the number multiplying a variable. In $6 a$ , the coefficient is 6.
A constant is a term with no variable, such as 9.
A factor is something being multiplied. In $5 (t + 2)$ , 5 and $t + 2$ are factors.
A power tells you repeated multiplication: $x^2$ means $\times x$ .

Like terms have the same variable part, such as $3 x$ and $8 x$ . You can collect like terms by adding or subtracting their coefficients.

Example

Collecting like terms

Start with $8 x + 5 x - 4$ .
The like terms are $8 x$ and $5 x$ because both have $x$ .
Add the coefficients:
$8 x + 5 x = 13 x$
The constant stays separate, so the simplified expression is $13 x - 4$ .

Expanding one bracket

Definition

Expand

To expand means to remove brackets by multiplying the factor outside the bracket by every term inside the bracket.

This rule is called the distributive law: the outside factor is distributed to each term inside the bracket.

Key Idea

Brackets mean multiply

In $a (b + c)$ , the outside factor $a$ multiplies both $b$ and $c$ , not just the first term.

The distributive law shows the outside factor multiplying every term inside the bracket.

Example

Expanding a simple bracket

Start with $6 (3 x + 4)$ . The outside factor is 6.

The 6 is distributed to both terms inside the bracket.

Multiply 6 by each term inside the bracket.
$\times 3x + 6 \times 4$
Work out each multiplication.
$6 (3 x + 4) = 18 x + 24$

Common Mistake

Only multiplying the first term

Do not leave the second term unchanged: $6 (3 x + 4)$ is not $18 x + 4$ . The 6 multiplies the 4 as well.

A common error is multiplying only the first term and forgetting the second term.

Expanding when there is a letter outside

The outside factor might include a letter, such as $3 p$ . Multiply it by each term in the bracket.

Example

Expanding with a letter outside

Start with $3 p (p - 5)$ . The outside factor is $3 p$ .

When the outside factor has a letter, it still multiplies every term inside the bracket.

Multiply $3 p$ by $p$ .
$3p \times p = 3p^2$
Multiply $3 p$ by $- 5$ .
$3p \times (-5) = -15p$
Put the two results together: $3p^2 - 15p$ .

Factorising: putting brackets back in

Definition

Factorise

To factorise means to write an expression using brackets by taking out a common factor. It is the reverse of expanding.

A common factor is a number, letter, or expression that is a factor of every term. The highest common factor, or HCF, is the largest common factor you can take out.

Example

Factorising a simple expression

Start with $8 r + 20$ .

Factorising takes the common factor out and places the remaining parts inside a bracket.

The HCF of 8 and 20 is 4.
Rewrite each term using 4.
$8 r + 20 = 4 (2 r) + 4 (5)$
Put the common factor outside the bracket.
$8 r + 20 = 4 (2 r + 5)$

Tip

Check by expanding

To check factorising, expand your answer. If $4 (2 r + 5)$ gives $8 r + 20$ , your brackets are correct.

Factorising fully with letters

To factorise fully means take out the biggest common factor, including any letters that appear in every term.

Example

Factorising fully

Start with $14a^2b + 21ab^2$ .

Factorising fully means taking out the full common factor shared by both terms.

The HCF of 14 and 21 is 7.
Both terms contain $a$ and $b$ , so the letter part of the common factor is $ab$ .
The full common factor is $7 ab$ . Divide each term by $7 ab$ .
$14a2b÷7ab=2a21ab2÷7ab=3b\begin{aligned} 14a^2b \div 7ab &= 2a\\ 21ab^2 \div 7ab &= 3b \end{aligned}$
Write the answer as $7 ab (2 a + 3 b)$ .

Common Mistake

Stopping too early

$14a^2b+21ab^2=7(2a^2b+3ab^2)$ is not fully factorised because $a$ and $b$ can still be taken out.

Expanding and simplifying

To simplify means to write an expression in a shorter form, usually by collecting like terms. For these questions, expand first, then collect like terms.

Example

Expand and simplify

Start with $5 (s + 3) + 2 (s - 4)$ .

Expanding both brackets first makes the like terms easy to collect.

Expand both brackets.
$5 (s + 3) + 2 (s - 4) = 5 s + 15 + 2 s - 8$
Collect the like terms.
$5 s + 2 s + 15 - 8 = 7 s + 7$

Common Mistake

Minus before a bracket

A minus sign before a bracket affects every term inside it. For example, $- 3 (y + 5)$ becomes $- 3 y - 15$ , not $- 3 y + 15$ .

Example

Subtracting a bracket

Start with $4 (2 y - 3) - 3 (y + 5)$ .

The minus before the second bracket means the outside factor is −3, so it multiplies both terms inside.

Expand both brackets. The second bracket has outside factor $- 3$ .
$4 (2 y - 3) - 3 (y + 5) = 8 y - 12 - 3 y - 15$
Collect like terms.
$8 y - 3 y - 12 - 15 = 5 y - 27$

Solving equations with brackets

An equation has an equals sign. To solve an equation means to find the value of the variable that makes both sides equal.

An inverse operation is an undo step. For example, division undoes multiplication, and subtraction undoes addition.

Tip

Divide first

If the whole bracket is multiplied by a number, dividing first is usually quicker than expanding.

Example

Solving a bracket equation

Start with $6 (d + 2) = 48$ . The bracket is multiplied by 6, so divide both sides by 6.
$d + 2 = 8$
Undo the plus 2 by subtracting 2 from both sides.
$d = 6$
Check in the original equation: $6 (6 + 2) = 48$ .

Example

When the answer is not a whole number

Start with $2 (v - 6) = 15$ . Divide both sides by 2.
$v - 6 = 7.5$
Undo the minus 6 by adding 6 to both sides.
$v = 13.5$
A decimal answer is fine if the division does not come out as a whole number.

Exam technique

In the exam

When expanding, make sure every term inside the bracket has been multiplied by the outside factor.
When factorising, look for the highest common factor and check by expanding your answer.
When solving equations, do the same inverse operation to both sides and check your value in the original equation.

Self review

Check yourself

Can you expand $5 (2 x - 3)$ without missing the negative sign?
Can you fully factorise $12a^2 + 18ab$ by taking out a number and a letter?
If $3 (k - 4) = 21$ , what is the first operation you would undo?

Expanding and Factorising

What you'll learn

The algebra basics

Expanding one bracket

Expanding when there is a letter outside

Factorising: putting brackets back in

Factorising fully with letters

Expanding and simplifying

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising

What you'll learn

The algebra basics

Expanding one bracket

Expanding when there is a letter outside

Factorising: putting brackets back in

Factorising fully with letters

Expanding and simplifying

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising

What you'll learn

The algebra basics

Expanding one bracket

Expanding when there is a letter outside

Factorising: putting brackets back in

Factorising fully with letters

Expanding and simplifying

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising Revision Guide

Expanding and Factorising

What you'll learn

The algebra basics

Expanding one bracket

Expanding when there is a letter outside

Factorising: putting brackets back in

Factorising fully with letters

Expanding and simplifying

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising Revision Guide