Expanding Brackets Worksheet, Questions and Revision

GCSE 1 - 3GCSE 4 - 5KS3
AQAEdexcelOCRWJEC

Specification Points Covered

Expanding Brackets

Expanding brackets is a key algebra skill that will be require to confidently tackle all sorts of algebra questions.

Make sure you are happy with the following topics before continuing:

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC

Expanding Single Brackets

The process by which we remove brackets is called expanding (or multiplying out) the brackets. This is the opposite process to factorising.

To expand 3(x+2) we need to multiply 3 by x and by 2

\textcolor{red}{3}(\textcolor{limegreen}{x}+\textcolor{blue}{2}) = (\textcolor{red}{3}\times \textcolor{limegreen}{x}) + (\textcolor{red}{3}\times \textcolor{blue}{2}) = \textcolor{red}{3}\textcolor{limegreen}{x} + \textcolor{purple}{6}

This can become harder as the terms get more tricky.

Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC
Expanding Single Brackets Positive Numbers

Example 1: Single Brackets

Expand the following, 2(3a+5)

[1 mark]

The green arrow shows the first calculation 2 \times 3a = 6a

The red arrow shows the second calculation 2 \times 5 = 10

This gives the final answer as 6a+10

Expanding Single Brackets Positive Numbers
Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC
Expanding Single Brackets With Negative Variables

Example 2: Single Brackets

Expand the following, -2y(2x-7y)

[2 marks]

The green arrow shows the first calculation -2y \times 2x = -4xy

The red arrow shows the second calculation -2y \times -7y = 14y^2

This gives the final answer as -4xy+14y^2

Expanding Single Brackets With Negative Variables
Level 1-3 GCSE KS3 AQA Edexcel OCR WJEC
Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

Expanding Double Brackets – Foil Method

When expanding double brackets, we need to multiply each of the things in the first bracket by each of the things in the second bracket. The  FOIL method is a way of ensuring this every time.

F – First ,   O – Outside,   I – Inner ,   L – Last

Expanding Double Brackets Foil Method Explained

Using the FOIL method will always give your answer in the same form, all you need to do is simplify by collecting the like terms.

\textcolor{red}{x^2} \textcolor{limegreen}{-2x}\textcolor{purple}{+5x}\textcolor{blue}{-10} = \textcolor{red}{x^2}  \textcolor{orange}{+ 3x} \textcolor{blue}{-10}

Level 4-5GCSEKS3AQAEdexcelOCRWJEC
Expanding Double Brackets Foil Method Example

Example 3: Expanding Double Brackets

Using FOIL expand and simplify the following (x + 3)(x - 4).

[3 marks]

Using the FOIL method we get

F  = \textcolor{red}{x \times x = x^2}

O  = \textcolor{limegreen}{x \times -4 = -4x}

I  =  \textcolor{purple}{3 \times x  = 3x}

L  = \textcolor{blue}{3 \times -4 = -12}

Expanding Double Brackets Foil Method Example

We must collect like terms to simplify our answer

\textcolor{red}{x^2} \textcolor{limegreen}{- 4x} \textcolor{purple}{+3x} \textcolor{blue}{-12} = \textcolor{red}{x^2}\textcolor{maroon}{ - x} \textcolor{blue}{- 12}.

Level 4-5 GCSE KS3 AQA Edexcel OCR WJEC

Example Questions

We need to multiply everything inside the bracket by 3xy, thus

 

3xy(x^2 +2x-8)  \\ = 3xy \times x^2 + 3xy \times 2x + 3xy \times (-8) \\ = 3x^3y + 6x^2y - 24xy

We need to multiply everything inside the bracket by 9pq, thus

 

9pq(2 - pq^2 - 7p^4) \\ =9pq \times 2 - 9pq \times pq^2 - 9pq \times 7p^4 \\ = 18pq -9p^{2}q^3 - 63p^{5}q

We need to make sure that we multiply everything in the left bracket by everything in the right bracket.

 

By using FOIL or another method of remembering to do every multiplication, we get

 

(y-3)(y-10) \\ = y\times y+y\times(-10)+(-3)\times y +(-3)\times(-10) \\ =y^2 -10y -3y +30

 

Then, collecting like terms we get the result of the expansion to be

 

y^2 -13y + 30

 

We need to make sure that we multiply everything in the left bracket by everything in the right bracket.

 

By using FOIL or some other method of remembering to do every multiplication, we get

 

(m+2n)(m-n) \\ = m\times m+m\times(-n)+2n\times m +2n\times(-n) \\ = m^2 -nm +2nm - 2n^2

 

Then, collecting like terms we get the result of the expansion to be

 

m^2 + nm - 2n^2

First, we can write this as two sets of brackets,

 

(2y^2+3x)^2=(2y^2+3x)(2y^2+3x)

 

By using FOIL and collecting like terms, we get

 

(2y^2+3x)(2y^2+3x) \\ =2y^2\times2y^2+2y^2\times3x+3x\times2y^2+3x\times3x \\ =4y^4+6xy^2+6xy^2+9x^2 \\ = 4y^4+12xy^2+9x^2

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