Expanding Triple Brackets

What you'll learn

How to expand brackets using the distributive law.
How to turn three linear brackets into a cubic expression.
How to deal with negatives, repeated brackets, and coefficients like $2 x$ or $3 x$ .
How to prove a “show that” expansion is true for all values of $x$ .

Key algebra words

Before triple brackets, make sure the basic language is secure.

Definition

Useful vocabulary

A term is a separate part of an expression, such as $3x^2$ , $- 5 x$ , or 7.
A coefficient is the number multiplying a variable. In $4 x$ , the coefficient is 4.
Like terms have the same variable part, such as $3x^2$ and $8x^2$ .
To simplify means to collect like terms and write the expression as neatly as possible.

Prerequisite 1: expanding one bracket

To expand a bracket means to multiply everything inside the bracket by what is outside it.

Definition

Expanding

Expanding removes brackets by multiplying each term inside the bracket by the factor outside or beside it.

For example, in $3 (x + 5)$ , the 3 multiplies both $x$ and 5.

Example

Expanding a single bracket

Expand and simplify $4 (x - 3) + 2 x$ .

Multiply 4 by each term inside the bracket:
$4 (x - 3) = 4 x - 12$
Put the rest of the expression back in:
$4 (x - 3) + 2 x = 4 x - 12 + 2 x$
Collect the like $x$ terms:
$4 x + 2 x - 12 = 6 x - 12$

Prerequisite 2: expanding two brackets

A pair of brackets such as $(x + 3) (x - 2)$ means every term in the first bracket must multiply every term in the second bracket.

Key Idea

Every term multiplies every term

When expanding brackets, do not just multiply the first terms. Each term in one bracket must be paired with each term in the other bracket.

Example

Expanding two brackets

Expand and simplify $(x + 4) (x - 3)$ .

Multiply the first $x$ by both terms in the second bracket:
$x(x-3)=x^2-3x$
Multiply the 4 by both terms in the second bracket:
$4 (x - 3) = 4 x - 12$
Combine the results:
$x+4)(x-3)=x^2-3x+4x-12$
Collect like terms:
$x^2-3x+4x-12=x^2+x-12$

Common Mistake

Only multiplying the outside terms

For $(x + 4) (x - 3)$ , writing $x^2-12$ is not enough. The middle terms $- 3 x$ and $4 x$ also appear, then combine to make $x$ .

What happens with triple brackets?

A triple bracket has three brackets multiplied together, for example $(x + 2) (x + 5) (x - 1)$ .

Each bracket is usually linear, meaning the highest power of $x$ inside it is $x^1$ . When you multiply three linear brackets, the answer is usually a cubic expression, meaning the highest power is $x^3$ .

Workflow for expanding triple brackets

Key Idea

The main method

Expand two brackets first to make a quadratic, then multiply that quadratic by the remaining bracket, then collect like terms.

You can choose any pair of brackets to expand first. The final answer will be the same, but some choices make the arithmetic easier.

Triple brackets where each bracket starts with $x$

Start with the most straightforward type: each bracket looks like $x$ plus or minus a number.

Example

Expanding three simple brackets

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

Choose two brackets to expand first. Here, expand the first two:
$x+3)(x+4)=x^2+4x+3x+12$
Collect like terms in the quadratic:
$x^2+4x+3x+12=x^2+7x+12$
Now multiply this quadratic by the remaining bracket:
$x^2+7x+12)(x+2)$
Multiply each term in the quadratic by $x$ :
$x(x^2+7x+12)=x^3+7x^2+12x$
Multiply each term in the quadratic by 2:
$2(x^2+7x+12)=2x^2+14x+24$
Add the two rows and collect like terms:
$x^3+7x^2+12x+2x^2+14x+24=x^3+9x^2+26x+24$

So:

x+3)(x+4)(x+2)=x^3+9x^2+26x+24

Tip

Check the shape of your answer

For three brackets each containing an $x$ term, expect an answer with powers going down: $x^3$ , then $x^2$ , then $x$ , then a constant.

Handling negative signs carefully

Negative signs are the main source of errors in this topic. Treat subtraction as adding a negative number.

For example, $x - 5$ means $x + (- 5)$ .

Example

Expanding with mixed signs

Expand and simplify $(x - 4) (x + 6) (x - 1)$ .

Expand the first two brackets:
$x-4)(x+6)=x^2+6x-4x-24$
Collect the middle terms:
$x^2+6x-4x-24=x^2+2x-24$
Multiply the quadratic by the remaining bracket:
$x^2+2x-24)(x-1)$
Multiply the quadratic by $x$ :
$x(x^2+2x-24)=x^3+2x^2-24x$
Multiply the quadratic by $- 1$ :
$1(x^2+2x-24)=-x^2-2x+24$
Add and collect like terms:
$x^3+2x^2-24x-x^2-2x+24=x^3+x^2-26x+24$

Common Mistake

Dropping the negative multiplier

In $x^2+2x-24)(x-1)$ , the second part is multiplying by $- 1$ , not by 1. This changes every sign in the quadratic.

Repeated brackets, such as $x-2)^2$

A squared bracket means the bracket is multiplied by itself.

So $x-2)^2$ means $(x - 2) (x - 2)$ .

Example

Expanding a repeated bracket

Expand and simplify $x+5)(x-2)^2$ .

Rewrite the squared bracket as two identical brackets:
$x+5)(x-2)^2=(x+5)(x-2)(x-2)$
Expand the repeated pair first:
$x-2)(x-2)=x^2-2x-2x+4$
Simplify the quadratic:
$x^2-2x-2x+4=x^2-4x+4$
Multiply by the remaining bracket:
$x+5)(x^2-4x+4)$
Multiply the quadratic by $x$ and by 5:
$x(x^2-4x+4)=x^3-4x^2+4x$ $5(x^2-4x+4)=5x^2-20x+20$
Add and collect like terms:
$x^3-4x^2+4x+5x^2-20x+20=x^3+x^2-16x+20$

Common Mistake

Do not square term by term

$x-2)^2$ is not $x^2+4$ . You must write it as $(x - 2) (x - 2)$ and expand properly.

Triple brackets with coefficients

Some brackets contain terms like $2 x$ , $3 x$ , or $5 x$ . The method is exactly the same, but the leading coefficient of the final $x^3$ term may not be 1.

Example

Expanding with coefficients

Expand and simplify $(2 x + 3) (x - 1) (3 x + 2)$ .

Expand the first two brackets:
$2x+3)(x-1)=2x^2-2x+3x-3$
Simplify the quadratic:
$2x^2-2x+3x-3=2x^2+x-3$
Multiply by the remaining bracket:
$2x^2+x-3)(3x+2)$
Multiply the quadratic by $3 x$ :
$3x(2x^2+x-3)=6x^3+3x^2-9x$
Multiply the quadratic by 2:
$2(2x^2+x-3)=4x^2+2x-6$
Add and collect like terms:
$6x^3+3x^2-9x+4x^2+2x-6=6x^3+7x^2-7x-6$

Tip

Leading term shortcut

To check the first term, multiply the $x$ parts from all three brackets. For $(2 x + 3) (x - 1) (3 x + 2)$ , the leading term is $2x \cdot x \cdot 3x=6x^3$ .

“Show that” questions

Sometimes you are asked to show that a triple bracket equals a given cubic expression for all values of $x$ .

Definition

Identity

An identity is an equation that is true for every allowed value of the variable. Expanding one side until it matches the other side proves the identity.

In these questions, do not substitute one value of $x$ . That only checks one case. Instead, expand the left-hand side and simplify it until it matches the right-hand side.

Example

Proving a given expansion

Show that $2x+1)(4x-3)(x-2)=8x^3-18x^2+x+6$ for all values of $x$ .

Expand the first two brackets on the left-hand side:
$2x+1)(4x-3)=8x^2-6x+4x-3$
Simplify:
$8x^2-6x+4x-3=8x^2-2x-3$
Multiply by the remaining bracket:
$8x^2-2x-3)(x-2)$
Multiply the quadratic by $x$ :
$x(8x^2-2x-3)=8x^3-2x^2-3x$
Multiply the quadratic by $- 2$ :
$2(8x^2-2x-3)=-16x^2+4x+6$
Add and collect like terms:
$8x^3-2x^2-3x-16x^2+4x+6=8x^3-18x^2+x+6$
The simplified left-hand side matches the right-hand side:
$2x+1)(4x-3)(x-2)=8x^3-18x^2+x+6$

A reliable layout for your working

For full marks, make your method easy to follow. A neat layout is:

Expand two brackets.
Simplify the quadratic.
Multiply by the third bracket.
Collect like terms.
Write the final expression in descending powers: $x^3$ , $x^2$ , $x$ , constant.

Exam technique

In the exam

Choose the easiest pair of brackets first, often a pair with small numbers or a repeated bracket.
Keep negative signs attached to their terms, especially when multiplying by something like $- 3$ or $- 2 x$ .
Check your final answer has been fully simplified and written in descending powers.

Self review

Check yourself

Can you expand $(x + 2) (x - 3)$ without missing the two middle terms?
If you see $x-4)^2$ , do you automatically rewrite it as $(x - 4) (x - 4)$ ?
In a “show that” question, can you explain why expanding proves the result for all values of $x$ ?

Expanding Triple Brackets

What you'll learn

How to expand brackets using the distributive law.
How to turn three linear brackets into a cubic expression.
How to deal with negatives, repeated brackets, and coefficients like $2 x$ or $3 x$ .
How to prove a “show that” expansion is true for all values of $x$ .

Key algebra words

Before triple brackets, make sure the basic language is secure.

Definition

Useful vocabulary

A term is a separate part of an expression, such as $3x^2$ , $- 5 x$ , or 7.
A coefficient is the number multiplying a variable. In $4 x$ , the coefficient is 4.
Like terms have the same variable part, such as $3x^2$ and $8x^2$ .
To simplify means to collect like terms and write the expression as neatly as possible.

Prerequisite 1: expanding one bracket

To expand a bracket means to multiply everything inside the bracket by what is outside it.

Definition

Expanding

Expanding removes brackets by multiplying each term inside the bracket by the factor outside or beside it.

For example, in $3 (x + 5)$ , the 3 multiplies both $x$ and 5.

Example

Expanding a single bracket

Expand and simplify $4 (x - 3) + 2 x$ .

Multiply 4 by each term inside the bracket:
$4 (x - 3) = 4 x - 12$
Put the rest of the expression back in:
$4 (x - 3) + 2 x = 4 x - 12 + 2 x$
Collect the like $x$ terms:
$4 x + 2 x - 12 = 6 x - 12$

Prerequisite 2: expanding two brackets

A pair of brackets such as $(x + 3) (x - 2)$ means every term in the first bracket must multiply every term in the second bracket.

Key Idea

Every term multiplies every term

When expanding brackets, do not just multiply the first terms. Each term in one bracket must be paired with each term in the other bracket.

Example

Expanding two brackets

Expand and simplify $(x + 4) (x - 3)$ .

Multiply the first $x$ by both terms in the second bracket:
$x(x-3)=x^2-3x$
Multiply the 4 by both terms in the second bracket:
$4 (x - 3) = 4 x - 12$
Combine the results:
$x+4)(x-3)=x^2-3x+4x-12$
Collect like terms:
$x^2-3x+4x-12=x^2+x-12$

Common Mistake

Only multiplying the outside terms

For $(x + 4) (x - 3)$ , writing $x^2-12$ is not enough. The middle terms $- 3 x$ and $4 x$ also appear, then combine to make $x$ .

What happens with triple brackets?

A triple bracket has three brackets multiplied together, for example $(x + 2) (x + 5) (x - 1)$ .

Workflow for expanding triple brackets

Key Idea

The main method

Expand two brackets first to make a quadratic, then multiply that quadratic by the remaining bracket, then collect like terms.

You can choose any pair of brackets to expand first. The final answer will be the same, but some choices make the arithmetic easier.

Triple brackets where each bracket starts with $x$

Start with the most straightforward type: each bracket looks like $x$ plus or minus a number.

Example

Expanding three simple brackets

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

Choose two brackets to expand first. Here, expand the first two:
$x+3)(x+4)=x^2+4x+3x+12$
Collect like terms in the quadratic:
$x^2+4x+3x+12=x^2+7x+12$
Now multiply this quadratic by the remaining bracket:
$x^2+7x+12)(x+2)$
Multiply each term in the quadratic by $x$ :
$x(x^2+7x+12)=x^3+7x^2+12x$
Multiply each term in the quadratic by 2:
$2(x^2+7x+12)=2x^2+14x+24$
Add the two rows and collect like terms:
$x^3+7x^2+12x+2x^2+14x+24=x^3+9x^2+26x+24$

So:

x+3)(x+4)(x+2)=x^3+9x^2+26x+24

Tip

Check the shape of your answer

For three brackets each containing an $x$ term, expect an answer with powers going down: $x^3$ , then $x^2$ , then $x$ , then a constant.

Handling negative signs carefully

Negative signs are the main source of errors in this topic. Treat subtraction as adding a negative number.

For example, $x - 5$ means $x + (- 5)$ .

Example

Expanding with mixed signs

Expand and simplify $(x - 4) (x + 6) (x - 1)$ .

Expand the first two brackets:
$x-4)(x+6)=x^2+6x-4x-24$
Collect the middle terms:
$x^2+6x-4x-24=x^2+2x-24$
Multiply the quadratic by the remaining bracket:
$x^2+2x-24)(x-1)$
Multiply the quadratic by $x$ :
$x(x^2+2x-24)=x^3+2x^2-24x$
Multiply the quadratic by $- 1$ :
$1(x^2+2x-24)=-x^2-2x+24$
Add and collect like terms:
$x^3+2x^2-24x-x^2-2x+24=x^3+x^2-26x+24$

Common Mistake

Dropping the negative multiplier

In $x^2+2x-24)(x-1)$ , the second part is multiplying by $- 1$ , not by 1. This changes every sign in the quadratic.

Repeated brackets, such as $x-2)^2$

A squared bracket means the bracket is multiplied by itself.

So $x-2)^2$ means $(x - 2) (x - 2)$ .

Example

Expanding a repeated bracket

Expand and simplify $x+5)(x-2)^2$ .

Rewrite the squared bracket as two identical brackets:
$x+5)(x-2)^2=(x+5)(x-2)(x-2)$
Expand the repeated pair first:
$x-2)(x-2)=x^2-2x-2x+4$
Simplify the quadratic:
$x^2-2x-2x+4=x^2-4x+4$
Multiply by the remaining bracket:
$x+5)(x^2-4x+4)$
Multiply the quadratic by $x$ and by 5:
$x(x^2-4x+4)=x^3-4x^2+4x$ $5(x^2-4x+4)=5x^2-20x+20$
Add and collect like terms:
$x^3-4x^2+4x+5x^2-20x+20=x^3+x^2-16x+20$

Common Mistake

Do not square term by term

$x-2)^2$ is not $x^2+4$ . You must write it as $(x - 2) (x - 2)$ and expand properly.

Triple brackets with coefficients

Some brackets contain terms like $2 x$ , $3 x$ , or $5 x$ . The method is exactly the same, but the leading coefficient of the final $x^3$ term may not be 1.

Example

Expanding with coefficients

Expand and simplify $(2 x + 3) (x - 1) (3 x + 2)$ .

Expand the first two brackets:
$2x+3)(x-1)=2x^2-2x+3x-3$
Simplify the quadratic:
$2x^2-2x+3x-3=2x^2+x-3$
Multiply by the remaining bracket:
$2x^2+x-3)(3x+2)$
Multiply the quadratic by $3 x$ :
$3x(2x^2+x-3)=6x^3+3x^2-9x$
Multiply the quadratic by 2:
$2(2x^2+x-3)=4x^2+2x-6$
Add and collect like terms:
$6x^3+3x^2-9x+4x^2+2x-6=6x^3+7x^2-7x-6$

Tip

Leading term shortcut

To check the first term, multiply the $x$ parts from all three brackets. For $(2 x + 3) (x - 1) (3 x + 2)$ , the leading term is $2x \cdot x \cdot 3x=6x^3$ .

“Show that” questions

Sometimes you are asked to show that a triple bracket equals a given cubic expression for all values of $x$ .

Definition

Identity

An identity is an equation that is true for every allowed value of the variable. Expanding one side until it matches the other side proves the identity.

In these questions, do not substitute one value of $x$ . That only checks one case. Instead, expand the left-hand side and simplify it until it matches the right-hand side.

Example

Proving a given expansion

Show that $2x+1)(4x-3)(x-2)=8x^3-18x^2+x+6$ for all values of $x$ .

Expand the first two brackets on the left-hand side:
$2x+1)(4x-3)=8x^2-6x+4x-3$
Simplify:
$8x^2-6x+4x-3=8x^2-2x-3$
Multiply by the remaining bracket:
$8x^2-2x-3)(x-2)$
Multiply the quadratic by $x$ :
$x(8x^2-2x-3)=8x^3-2x^2-3x$
Multiply the quadratic by $- 2$ :
$2(8x^2-2x-3)=-16x^2+4x+6$
Add and collect like terms:
$8x^3-2x^2-3x-16x^2+4x+6=8x^3-18x^2+x+6$
The simplified left-hand side matches the right-hand side:
$2x+1)(4x-3)(x-2)=8x^3-18x^2+x+6$

A reliable layout for your working

For full marks, make your method easy to follow. A neat layout is:

Expand two brackets.
Simplify the quadratic.
Multiply by the third bracket.
Collect like terms.
Write the final expression in descending powers: $x^3$ , $x^2$ , $x$ , constant.

Exam technique

In the exam

Choose the easiest pair of brackets first, often a pair with small numbers or a repeated bracket.
Keep negative signs attached to their terms, especially when multiplying by something like $- 3$ or $- 2 x$ .
Check your final answer has been fully simplified and written in descending powers.

Self review

Check yourself

Can you expand $(x + 2) (x - 3)$ without missing the two middle terms?
If you see $x-4)^2$ , do you automatically rewrite it as $(x - 4) (x - 4)$ ?
In a “show that” question, can you explain why expanding proves the result for all values of $x$ ?

Expanding Triple Brackets

What you'll learn

Key algebra words

Prerequisite 1: expanding one bracket

Prerequisite 2: expanding two brackets

What happens with triple brackets?

Triple brackets where each bracket starts with $x$

Handling negative signs carefully

Repeated brackets, such as $x-2)^2$

Triple brackets with coefficients

“Show that” questions

A reliable layout for your working

Recap questions

You've reached the end

Expanding Triple Brackets

What you'll learn

Key algebra words

Prerequisite 1: expanding one bracket

Prerequisite 2: expanding two brackets

What happens with triple brackets?

Triple brackets where each bracket starts with $x$

Handling negative signs carefully

Repeated brackets, such as $x-2)^2$

Triple brackets with coefficients

“Show that” questions

A reliable layout for your working

Recap questions

You've reached the end

Expanding Triple Brackets

What you'll learn

Key algebra words

Prerequisite 1: expanding one bracket

Prerequisite 2: expanding two brackets

What happens with triple brackets?

Triple brackets where each bracket starts with xxx

Handling negative signs carefully

Repeated brackets, such as (x−2)2(x-2)^2(x−2)2

Triple brackets with coefficients

“Show that” questions

A reliable layout for your working

Recap questions

You've reached the end

Expanding Triple Brackets Revision Guide

Expanding Triple Brackets

What you'll learn

Key algebra words

Prerequisite 1: expanding one bracket

Prerequisite 2: expanding two brackets

What happens with triple brackets?

Triple brackets where each bracket starts with xxx

Handling negative signs carefully

Repeated brackets, such as (x−2)2(x-2)^2(x−2)2

Triple brackets with coefficients

“Show that” questions

A reliable layout for your working

Recap questions

You've reached the end

Expanding Triple Brackets Revision Guide

Triple brackets where each bracket starts with $x$

Repeated brackets, such as $x-2)^2$

Triple brackets where each bracket starts with $x$

Repeated brackets, such as $x-2)^2$