Expanding Brackets

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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:

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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.

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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

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.

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

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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$

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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$

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

We must collect like terms to simplify our answer

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

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Expanding Brackets Example Questions

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

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

Gold Standard Education

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

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

Gold Standard Education

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

$y^2 -13y + 30$

Gold Standard Education

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

$m^2 + nm - 2n^2$

Gold Standard Education

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

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

Gold Standard Education

Expanding Brackets Worksheet and Example Questions

(NEW) Expanding Single Brackets Exam Style Questions - MME

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Expanding Brackets Drill Questions

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Multiplying Out Brackets

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