Expanding Triple Brackets
x

Revision notes for Edexcel GCSE Maths Expanding Triple Brackets. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Expanding Triple Brackets

What you'll learn

  • How to expand two brackets confidently before moving to three.
  • How to keep signs organised when brackets contain negatives.
  • How to handle squared brackets like (x−2)2\left(x - 2\right)^2(x−2)2.
  • How to answer “show that” questions using clear algebra steps.

The basics: expanding and simplifying

Definition

Expand and simplify

To expand means to remove brackets by multiplying everything out. To simplify means to collect like terms, which are terms with the same variable part, such as 3x23x^23x2 and −5x2-5x^2−5x2.

Before triple brackets, you need to be comfortable multiplying a single term into a bracket and collecting like terms.

For example:

2x(x+4)=2x2+8x2x(x + 4) = 2x^2 + 8x2x(x+4)=2x2+8x

The term 2x2x2x multiplies both terms inside the bracket.

Example

Collecting like terms

Simplify 4x2+7x−2x2+5x−34x^2 + 7x - 2x^2 + 5x - 34x2+7x−2x2+5x−3.

Like terms are grouped by matching variable parts before being combined.

  1. Group the like terms together:

    4x2−2x2+7x+5x−34x^2 - 2x^2 + 7x + 5x - 34x2−2x2+7x+5x−3

  2. Combine the x2x^2x2 terms, then the xxx terms:

    2x2+12x−32x^2 + 12x - 32x2+12x−3

  3. The simplified expression is:

    2x2+12x−32x^2 + 12x - 32x2+12x−3
Common Mistake

Combining unlike terms

You cannot combine x2x^2x2 terms with xxx terms. For example, 3x2+4x3x^2 + 4x3x2+4x does not become 7x37x^37x3.

Revising double brackets

A double bracket expression has two brackets multiplied together, such as (x+3)(x+5)\left(x + 3\right)\left(x + 5\right)(x+3)(x+5).

Key Idea

Every term multiplies every term

When expanding brackets, each term in one bracket must multiply each term in the other bracket.

A neat method is FOIL:

  • First terms
  • Outside terms
  • Inside terms
  • Last terms
Example

Expanding two brackets

Expand and simplify (x+6)(x−2)\left(x + 6\right)\left(x - 2\right)(x+6)(x−2).

A FOIL diagram shows the four products needed when expanding two brackets.

  1. Multiply the first terms:

    x×x=x2x \times x = x^2x×x=x2

  2. Multiply the outside terms:

    x×−2=−2xx \times -2 = -2xx×−2=−2x

  3. Multiply the inside terms:

    6×x=6x6 \times x = 6x6×x=6x

  4. Multiply the last terms:

    6×−2=−126 \times -2 = -126×−2=−12

  5. Collect the terms:

    x2−2x+6x−12=x2+4x−12x^2 - 2x + 6x - 12 = x^2 + 4x - 12x2−2x+6x−12=x2+4x−12
Tip

Sign check

When multiplying signs: positive times negative is negative, and negative times negative is positive.

Expanding triple brackets

A triple bracket expression has three brackets multiplied together, such as (x+1)(x+2)(x+3)\left(x + 1\right)\left(x + 2\right)\left(x + 3\right)(x+1)(x+2)(x+3).

You do not expand all three at once. The safest method is:

  1. Expand two brackets first.
  2. Simplify them.
  3. Multiply the result by the remaining bracket.
  4. Simplify again.
Key Idea

Do it in stages

Triple brackets are just double brackets followed by one more bracket expansion.

Example

Expanding three simple brackets

Expand and simplify (x+3)(x+5)(x+2)\left(x + 3\right)\left(x + 5\right)\left(x + 2\right)(x+3)(x+5)(x+2).

Triple brackets are expanded in stages: first make a quadratic, then multiply by the remaining bracket.

  1. Expand the first two brackets:

    (x+3)(x+5)=x2+5x+3x+15\left(x + 3\right)\left(x + 5\right) = x^2 + 5x + 3x + 15(x+3)(x+5)=x2+5x+3x+15

  2. Simplify the double bracket result:

    x2+8x+15x^2 + 8x + 15x2+8x+15

  3. Now multiply this by the remaining bracket:

    (x2+8x+15)(x+2)\left(x^2 + 8x + 15\right)\left(x + 2\right)(x2+8x+15)(x+2)

  4. Multiply each term in x2+8x+15x^2 + 8x + 15x2+8x+15 by each term in x+2x + 2x+2:

    x3+2x2+8x2+16x+15x+30x^3 + 2x^2 + 8x^2 + 16x + 15x + 30x3+2x2+8x2+16x+15x+30

  5. Collect like terms:

    x3+10x2+31x+30x^3 + 10x^2 + 31x + 30x3+10x2+31x+30

Choosing which two brackets to expand first

You can choose any pair of brackets first, but it is usually easiest to start with two simple brackets like (x+a)(x+b)\left(x + a\right)\left(x + b\right)(x+a)(x+b).

Example

Choosing a helpful pair

Expand and simplify (x−4)(x+2)(x+7)\left(x - 4\right)\left(x + 2\right)\left(x + 7\right)(x−4)(x+2)(x+7).

Choosing two simple brackets first turns the triple product into a quadratic times a linear bracket.

  1. Start with the first two brackets:

    (x−4)(x+2)\left(x - 4\right)\left(x + 2\right)(x−4)(x+2)

  2. Expand and simplify:

    x2+2x−4x−8=x2−2x−8x^2 + 2x - 4x - 8 = x^2 - 2x - 8x2+2x−4x−8=x2−2x−8

  3. Multiply by the third bracket:

    (x2−2x−8)(x+7)\left(x^2 - 2x - 8\right)\left(x + 7\right)(x2−2x−8)(x+7)

  4. Expand term by term:

    x3+7x2−2x2−14x−8x−56x^3 + 7x^2 - 2x^2 - 14x - 8x - 56x3+7x2−2x2−14x−8x−56

  5. Collect like terms:

    x3+5x2−22x−56x^3 + 5x^2 - 22x - 56x3+5x2−22x−56
Common Mistake

Forgetting the third bracket

After expanding two brackets, you are not finished. You must still multiply the quadratic by the remaining bracket.

When brackets include coefficients

Definition

Coefficient

A coefficient is the number multiplying a variable. In 3x3x3x, the coefficient of xxx is 3.

Some triple bracket questions include terms like 2x−12x - 12x−1 or 3x+43x + 43x+4. The method is the same, but the arithmetic needs more care.

Example

Expanding with coefficients

Expand and simplify (2x+3)(x−1)(x+4)\left(2x + 3\right)\left(x - 1\right)\left(x + 4\right)(2x+3)(x−1)(x+4).

With coefficients, each term in the linear bracket still multiplies every term in the quadratic.

  1. Expand the two simpler brackets first:

    (x−1)(x+4)\left(x - 1\right)\left(x + 4\right)(x−1)(x+4)

  2. Simplify the result:

    x2+4x−x−4=x2+3x−4x^2 + 4x - x - 4 = x^2 + 3x - 4x2+4x−x−4=x2+3x−4

  3. Now multiply by the remaining bracket:

    (2x+3)(x2+3x−4)\left(2x + 3\right)\left(x^2 + 3x - 4\right)(2x+3)(x2+3x−4)

  4. Multiply 2x2x2x by every term in the quadratic:

    2x3+6x2−8x2x^3 + 6x^2 - 8x2x3+6x2−8x

  5. Multiply 3 by every term in the quadratic:

    3x2+9x−123x^2 + 9x - 123x2+9x−12

  6. Add and collect like terms:

    2x3+9x2+x−122x^3 + 9x^2 + x - 122x3+9x2+x−12
Tip

Highest power check

When multiplying three linear brackets, your answer should usually start with an x3x^3x3 term. If it only has x2x^2x2, something has been missed.

Squared brackets in triple bracket questions

Definition

Squared bracket

A squared bracket such as (x−3)2\left(x - 3\right)^2(x−3)2 means the bracket is multiplied by itself: (x−3)(x−3)\left(x - 3\right)\left(x - 3\right)(x−3)(x−3).

Do not square each term separately. For example, (x+5)2\left(x + 5\right)^2(x+5)2 is not x2+25x^2 + 25x2+25.

Example

Using a squared bracket

Expand and simplify (x+4)(x−2)2\left(x + 4\right)\left(x - 2\right)^2(x+4)(x−2)2.

A squared bracket means the same bracket appears twice, including the middle terms when expanded.

  1. Rewrite the squared bracket:

    (x+4)(x−2)(x−2)\left(x + 4\right)\left(x - 2\right)\left(x - 2\right)(x+4)(x−2)(x−2)

  2. Expand the repeated brackets first:

    (x−2)(x−2)=x2−2x−2x+4\left(x - 2\right)\left(x - 2\right) = x^2 - 2x - 2x + 4(x−2)(x−2)=x2−2x−2x+4

  3. Simplify:

    x2−4x+4x^2 - 4x + 4x2−4x+4

  4. Multiply by the remaining bracket:

    (x+4)(x2−4x+4)\left(x + 4\right)\left(x^2 - 4x + 4\right)(x+4)(x2−4x+4)

  5. Expand and collect:

    x3−4x2+4x+4x2−16x+16=x3−12x+16x^3 - 4x^2 + 4x + 4x^2 - 16x + 16 = x^3 - 12x + 16x3−4x2+4x+4x2−16x+16=x3−12x+16
Common Mistake

Squaring only the ends

Remember that (x−2)2\left(x - 2\right)^2(x−2)2 gives three terms after expansion, not just x2+4x^2 + 4x2+4. The middle term matters.

“Show that” questions

Definition

Identity

An identity is an equation that is true for all values of the variable. In a “show that” question, you prove both sides match by expanding and simplifying one side.

For these, expand the bracket side carefully until it matches the expression given.

Example

Showing an identity

Show that (2x−1)(x+3)(4x−5)=8x3+10x2−43x+15\left(2x - 1\right)\left(x + 3\right)\left(4x - 5\right) = 8x^3 + 10x^2 - 43x + 15(2x−1)(x+3)(4x−5)=8x3+10x2−43x+15 for all values of xxx.

For a show-that question, expand the left-hand side until it can be compared with the target expression.

  1. Expand two brackets first:

    (x+3)(4x−5)\left(x + 3\right)\left(4x - 5\right)(x+3)(4x−5)

  2. Simplify the result:

    4x2−5x+12x−15=4x2+7x−154x^2 - 5x + 12x - 15 = 4x^2 + 7x - 154x2−5x+12x−15=4x2+7x−15

  3. Multiply by the remaining bracket:

    (2x−1)(4x2+7x−15)\left(2x - 1\right)\left(4x^2 + 7x - 15\right)(2x−1)(4x2+7x−15)

  4. Expand using 2x2x2x first, then −1-1−1:

    8x3+14x2−30x−4x2−7x+158x^3 + 14x^2 - 30x - 4x^2 - 7x + 158x3+14x2−30x−4x2−7x+15

  5. Collect like terms:

    8x3+10x2−37x+158x^3 + 10x^2 - 37x + 158x3+10x2−37x+15
Common Mistake

Check the final expression

If your simplified answer does not match the expression you were asked to show, do not just write “shown”. Go back and check signs and arithmetic carefully.

A quick layout method

For longer expansions, you can write the multiplication in two chunks:

  • Multiply the first term of the linear bracket by the whole quadratic.
  • Multiply the second term of the linear bracket by the whole quadratic.
  • Add the two results.

This keeps your working neat and helps you avoid missing terms.

Example

Neat chunking method

Expand and simplify (3x−2)(x2+5x−6)\left(3x - 2\right)\left(x^2 + 5x - 6\right)(3x−2)(x2+5x−6).

The chunking method separates the multiplication by 3x and by -2 before collecting terms.

  1. Multiply 3x3x3x by the quadratic:

    3x3+15x2−18x3x^3 + 15x^2 - 18x3x3+15x2−18x

  2. Multiply −2-2−2 by the quadratic:

    −2x2−10x+12-2x^2 - 10x + 12−2x2−10x+12

  3. Add the two lines together:

    3x3+15x2−18x−2x2−10x+123x^3 + 15x^2 - 18x - 2x^2 - 10x + 123x3+15x2−18x−2x2−10x+12

  4. Collect like terms:

    3x3+13x2−28x+123x^3 + 13x^2 - 28x + 123x3+13x2−28x+12
Exam technique

In the exam

  1. Expand only two brackets first, then clearly multiply by the third.
  2. Keep negative signs attached to their terms, especially when multiplying by a negative number.
  3. Check that your final answer is in descending powers: x3x^3x3, then x2x^2x2, then xxx, then the constant.
Self review

Check yourself

  • Can you explain why (x−3)2\left(x - 3\right)^2(x−3)2 is not the same as x2−9x^2 - 9x2−9?
  • When expanding three linear brackets, what power of xxx should the first term usually have?
  • Can you spot where like terms should be collected in a long expansion?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

You've reached the end

Test yourself on this topic, or move on to the next guide.

Practice questionsTake a quick quiz on this topicFlashcardsSelf-test with active recall
Parallel and Perpendicular LinesUp next

How was this guide?

Expanding Triple Brackets Revision Guide

  1. GCSE
  2. /Maths
  3. /Expanding Triple Brackets