Expanding and Factorising

What you'll learn

How to expand a single bracket by multiplying every term inside it.
How to collect like terms after expanding more than one bracket.
How to factorise by taking out the highest common factor.
How to solve simple equations that contain brackets.

Algebra building blocks

Before expanding and factorising, you need to feel confident with the basic parts of an algebra expression.

Definition

Key algebra words

A variable is a letter that represents a number, such as $x$ , $y$ or $a$ .
A term is one part of an expression, separated by plus or minus signs. For example, in $5 x + 3$ , the terms are $5 x$ and 3.
A coefficient is the number multiplying a variable. In $7 x$ , the coefficient is 7.
Like terms have the same variable part, such as $4 x$ and $9 x$ , or $3a^2$ and $8a^2$ .

You can only collect like terms. For example, $6 x$ and $2 x$ can combine, but $6 x$ and $2 y$ cannot.

Example

Collecting like terms

Simplify $7 x + 5 + 4 x - 2 - 3 x$ .

Identify the like terms with $x$ : $7 x$ , $4 x$ and $- 3 x$ .
Combine the $x$ terms:
$7 x + 4 x - 3 x = 8 x$
Combine the number terms: 5 - 2 = 3.
Write the simplified expression as $8 x + 3$ .

Tip

Check like terms carefully

The variable part must match exactly. $x$ and $x^2$ are not like terms, because $x^2$ means $\times x$ .

Expanding one bracket

Definition

Expanding

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

The main rule is called the distributive law. It means the outside factor is shared across everything inside the bracket:

k (a + b) = k a + k b

k (a - b) = k a - k b

Expanding can be thought of as an area split into two parts: the outside factor is the common height, and each term inside the bracket is a width. The total area is the sum of the two smaller areas.

Area model showing a bracket being expanded into two rectangle areas

Key Idea

Multiply every term

When expanding a bracket, the outside multiplier must multiply every term inside the bracket, including negative terms.

Example

Expanding a number outside a bracket

Expand $8 (4 s - 3)$ .

The outside number is 8, so 8 multiplies both $4 s$ and -3.
Multiply the first term:
$\times 4s = 32s$
Multiply the second term:
$\times (-3) = -24$
Put the two results together: $8 (4 s - 3) = 32 s - 24$ .

Sometimes the term outside the bracket contains a variable as well.

Example

Expanding with a variable outside

Expand $5 x (3 x - 2)$ .

Multiply $5 x$ by $3 x$ :
$5x \times 3x = 15x^2$
Multiply $5 x$ by -2:
$5x \times (-2) = -10x$
Write the expanded expression: $5x(3x - 2) = 15x^2 - 10x$ .

Common Mistake

Only multiplying the first term

Do not write $8 (4 s - 3) = 32 s - 3$ . The 8 must also multiply the -3.

Expanding and simplifying

Definition

Simplifying

To simplify means to write an expression in a shorter or cleaner form, usually by collecting like terms.

When you see two brackets, expand each bracket first. Then collect like terms.

Example

Expanding and simplifying with addition

Expand and simplify $6 (g + 4) + 2 (g - 7)$ .

Expand the first bracket:
$6 (g + 4) = 6 g + 24$
Expand the second bracket:
$2 (g - 7) = 2 g - 14$
Put the expanded parts together:
$6 g + 24 + 2 g - 14$
Collect like terms: $6 g + 2 g = 8 g$ , and 24 - 14 = 10.
The simplified answer is $8 g + 10$ .

A minus sign before a bracket is especially important. It changes the signs of the terms in that bracket when expanded.

Example

Expanding and simplifying with subtraction

Expand and simplify $4 (3 y - 5) - 2 (5 y + 1)$ .

Expand the first bracket:
$4 (3 y - 5) = 12 y - 20$
Treat the subtraction as multiplying the second bracket by -2:
$- 2 (5 y + 1) = - 10 y - 2$
Put everything together:
$12 y - 20 - 10 y - 2$
Collect like terms: $12 y - 10 y = 2 y$ , and -20 - 2 = -22.
The simplified answer is $2 y - 22$ .

Common Mistake

Forgetting the second sign change

In $- 2 (5 y + 1)$ , both terms change: $5 y$ becomes $- 10 y$ and 1 becomes -2.

Factorising with a common factor

Factorising is the opposite of expanding.

For example:

6 (3 x + 4) = 18 x + 24

So the reverse is:

18 x + 24 = 6 (3 x + 4)

Definition

Factorising fully

To factorise means to rewrite an expression as multiplication using brackets. To factorise fully, you take out the highest common factor from every term.

The highest common factor, or HCF, is the biggest factor shared by the terms. It can include numbers and variables.

Example

Factorising a simple expression

Factorise $20 y + 35$ .

Find the HCF of the number parts 20 and 35, which is 5.
Divide each term by 5: $20y \div 5 = 4y$ and $35 \div 5 = 7$ .
Put the HCF outside the bracket:
$20 y + 35 = 5 (4 y + 7)$

When variables are involved, take out the variable part that appears in every term.

Example

Factorising fully with variables

Factorise fully $12m^2n + 18mn^2$ .

Find the HCF of the number parts 12 and 18, which is 6.
Look at the variables. Both terms contain at least one $m$ and one $n$ , so the variable part of the HCF is $mn$ .
The full HCF is $6 mn$ .
Divide each term by $6 mn$ : $12m2n÷6mn=2m12m^2n \div 6mn = 2m$ and $18mn2÷6mn=3n18mn^2 \div 6mn = 3n$ .
Write the factorised expression:
$12m^2n + 18mn^2 = 6mn(2m + 3n)$

Tip

Check by expanding back

After factorising, quickly expand your answer. If it returns to the original expression, your factorisation is correct.

Common Mistake

Not factorising fully

For $18 x + 24$ , writing $3 (6 x + 8)$ is factorising, but not fully. The HCF is 6, so the fully factorised answer is $6 (3 x + 4)$ .

Solving equations with brackets

Definition

Solving an equation

To solve an equation means to find the value of the variable that makes the equation true.

An inverse operation is an operation that undoes another one. Addition and subtraction undo each other; multiplication and division undo each other.

For equations like $5 (b + 4) = 45$ , it is often quickest to divide first, then deal with the bracket.

Example

Solving an equation with a bracket

Solve $5 (b + 4) = 45$ .

The bracket is being multiplied by 5, so divide both sides by 5:
$b + 4 = 9$
Subtract 4 from both sides:
$b = 5$
Check by substituting: $5 (5 + 4) = 45$ , so the solution works.

Sometimes the answer is not a whole number. That is fine.

Example

Solving when the answer is a decimal

Solve $4 (c - 3) = 18$ .

Divide both sides by 4:
$\frac{18}{4}$
Simplify $184\frac{18}{4}$ to $4.5$ .
Add 3 to both sides: $c = 7.5$ .

Exam technique

In the exam

Read the command word carefully: expand means remove brackets, factorise means create brackets, and solve means find the unknown.
When expanding, multiply the outside factor by every term inside the bracket, including negative terms.
When factorising fully, take out the HCF of the numbers and the variables, then expand your answer to check it.

Self review

Check yourself

If you expand $3 (2 x - 5)$ , what must the 3 multiply?
What is the HCF of the terms in $12a^2b + 18ab^2$ ?
When solving $4 (y + 2) = 28$ , what operation should you undo first?

Expanding and Factorising

What you'll learn

How to expand a single bracket by multiplying every term inside it.
How to collect like terms after expanding more than one bracket.
How to factorise by taking out the highest common factor.
How to solve simple equations that contain brackets.

Algebra building blocks

Before expanding and factorising, you need to feel confident with the basic parts of an algebra expression.

Definition

Key algebra words

A variable is a letter that represents a number, such as $x$ , $y$ or $a$ .
A term is one part of an expression, separated by plus or minus signs. For example, in $5 x + 3$ , the terms are $5 x$ and 3.
A coefficient is the number multiplying a variable. In $7 x$ , the coefficient is 7.
Like terms have the same variable part, such as $4 x$ and $9 x$ , or $3a^2$ and $8a^2$ .

You can only collect like terms. For example, $6 x$ and $2 x$ can combine, but $6 x$ and $2 y$ cannot.

Example

Collecting like terms

Simplify $7 x + 5 + 4 x - 2 - 3 x$ .

Identify the like terms with $x$ : $7 x$ , $4 x$ and $- 3 x$ .
Combine the $x$ terms:
$7 x + 4 x - 3 x = 8 x$
Combine the number terms: 5 - 2 = 3.
Write the simplified expression as $8 x + 3$ .

Tip

Check like terms carefully

The variable part must match exactly. $x$ and $x^2$ are not like terms, because $x^2$ means $\times x$ .

Expanding one bracket

Definition

Expanding

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

The main rule is called the distributive law. It means the outside factor is shared across everything inside the bracket:

k (a + b) = k a + k b

k (a - b) = k a - k b

Expanding can be thought of as an area split into two parts: the outside factor is the common height, and each term inside the bracket is a width. The total area is the sum of the two smaller areas.

Area model showing a bracket being expanded into two rectangle areas

Key Idea

Multiply every term

When expanding a bracket, the outside multiplier must multiply every term inside the bracket, including negative terms.

Example

Expanding a number outside a bracket

Expand $8 (4 s - 3)$ .

The outside number is 8, so 8 multiplies both $4 s$ and -3.
Multiply the first term:
$\times 4s = 32s$
Multiply the second term:
$\times (-3) = -24$
Put the two results together: $8 (4 s - 3) = 32 s - 24$ .

Sometimes the term outside the bracket contains a variable as well.

Example

Expanding with a variable outside

Expand $5 x (3 x - 2)$ .

Multiply $5 x$ by $3 x$ :
$5x \times 3x = 15x^2$
Multiply $5 x$ by -2:
$5x \times (-2) = -10x$
Write the expanded expression: $5x(3x - 2) = 15x^2 - 10x$ .

Common Mistake

Only multiplying the first term

Do not write $8 (4 s - 3) = 32 s - 3$ . The 8 must also multiply the -3.

Expanding and simplifying

Definition

Simplifying

To simplify means to write an expression in a shorter or cleaner form, usually by collecting like terms.

When you see two brackets, expand each bracket first. Then collect like terms.

Example

Expanding and simplifying with addition

Expand and simplify $6 (g + 4) + 2 (g - 7)$ .

Expand the first bracket:
$6 (g + 4) = 6 g + 24$
Expand the second bracket:
$2 (g - 7) = 2 g - 14$
Put the expanded parts together:
$6 g + 24 + 2 g - 14$
Collect like terms: $6 g + 2 g = 8 g$ , and 24 - 14 = 10.
The simplified answer is $8 g + 10$ .

A minus sign before a bracket is especially important. It changes the signs of the terms in that bracket when expanded.

Example

Expanding and simplifying with subtraction

Expand and simplify $4 (3 y - 5) - 2 (5 y + 1)$ .

Expand the first bracket:
$4 (3 y - 5) = 12 y - 20$
Treat the subtraction as multiplying the second bracket by -2:
$- 2 (5 y + 1) = - 10 y - 2$
Put everything together:
$12 y - 20 - 10 y - 2$
Collect like terms: $12 y - 10 y = 2 y$ , and -20 - 2 = -22.
The simplified answer is $2 y - 22$ .

Common Mistake

Forgetting the second sign change

In $- 2 (5 y + 1)$ , both terms change: $5 y$ becomes $- 10 y$ and 1 becomes -2.

Factorising with a common factor

Factorising is the opposite of expanding.

For example:

6 (3 x + 4) = 18 x + 24

So the reverse is:

18 x + 24 = 6 (3 x + 4)

Definition

Factorising fully

To factorise means to rewrite an expression as multiplication using brackets. To factorise fully, you take out the highest common factor from every term.

The highest common factor, or HCF, is the biggest factor shared by the terms. It can include numbers and variables.

Example

Factorising a simple expression

Factorise $20 y + 35$ .

Find the HCF of the number parts 20 and 35, which is 5.
Divide each term by 5: $20y \div 5 = 4y$ and $35 \div 5 = 7$ .
Put the HCF outside the bracket:
$20 y + 35 = 5 (4 y + 7)$

When variables are involved, take out the variable part that appears in every term.

Example

Factorising fully with variables

Factorise fully $12m^2n + 18mn^2$ .

Find the HCF of the number parts 12 and 18, which is 6.
Look at the variables. Both terms contain at least one $m$ and one $n$ , so the variable part of the HCF is $mn$ .
The full HCF is $6 mn$ .
Divide each term by $6 mn$ : $12m2n÷6mn=2m12m^2n \div 6mn = 2m$ and $18mn2÷6mn=3n18mn^2 \div 6mn = 3n$ .
Write the factorised expression:
$12m^2n + 18mn^2 = 6mn(2m + 3n)$

Tip

Check by expanding back

After factorising, quickly expand your answer. If it returns to the original expression, your factorisation is correct.

Common Mistake

Not factorising fully

For $18 x + 24$ , writing $3 (6 x + 8)$ is factorising, but not fully. The HCF is 6, so the fully factorised answer is $6 (3 x + 4)$ .

Solving equations with brackets

Definition

Solving an equation

To solve an equation means to find the value of the variable that makes the equation true.

An inverse operation is an operation that undoes another one. Addition and subtraction undo each other; multiplication and division undo each other.

For equations like $5 (b + 4) = 45$ , it is often quickest to divide first, then deal with the bracket.

Example

Solving an equation with a bracket

Solve $5 (b + 4) = 45$ .

The bracket is being multiplied by 5, so divide both sides by 5:
$b + 4 = 9$
Subtract 4 from both sides:
$b = 5$
Check by substituting: $5 (5 + 4) = 45$ , so the solution works.

Sometimes the answer is not a whole number. That is fine.

Example

Solving when the answer is a decimal

Solve $4 (c - 3) = 18$ .

Divide both sides by 4:
$\frac{18}{4}$
Simplify $184\frac{18}{4}$ to $4.5$ .
Add 3 to both sides: $c = 7.5$ .

Exam technique

In the exam

Read the command word carefully: expand means remove brackets, factorise means create brackets, and solve means find the unknown.
When expanding, multiply the outside factor by every term inside the bracket, including negative terms.
When factorising fully, take out the HCF of the numbers and the variables, then expand your answer to check it.

Self review

Check yourself

If you expand $3 (2 x - 5)$ , what must the 3 multiply?
What is the HCF of the terms in $12a^2b + 18ab^2$ ?
When solving $4 (y + 2) = 28$ , what operation should you undo first?

Expanding and Factorising

What you'll learn

Algebra building blocks

Expanding one bracket

Expanding and simplifying

Factorising with a common factor

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising

What you'll learn

Algebra building blocks

Expanding one bracket

Expanding and simplifying

Factorising with a common factor

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising

What you'll learn

Algebra building blocks

Expanding one bracket

Expanding and simplifying

Factorising with a common factor

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising Revision Guide

Expanding and Factorising

What you'll learn

Algebra building blocks

Expanding one bracket

Expanding and simplifying

Factorising with a common factor

Solving equations with brackets

Recap questions

You've reached the end

Expanding and Factorising Revision Guide