# Expanding Brackets

## Expanding Brackets Revision

**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:

**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.

**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}

**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

**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

**Example 3: Expanding Double Brackets**

Using **FOI****L** expand and simplify the following (x + 3)(x - 4).

**[3 marks]**

Using the **FOI****L **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}.

## Expanding Brackets Example Questions

**Question 1:** Expand 3xy(x^2 +2x-8)

**[2 marks]**

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

**Question 2:** Expand 9pq\left(2 - pq^2 - 7p^4\right)

**[2 marks]**

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

**Question 3: **Expand and simplify (y-3)(y-10)

**[3 marks]**

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

**Question 4:** Expand and simplify (m + 2n)(m - n)

**[3 marks]**

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

**Question 5:** Expand and simplify (2y^2+3x)^2

**[3 marks]**

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