### Expanding Brackets and Factorising (Economics)

#### Rules for Expanding Brackets

Expanding brackets, or multiplying out, involves multiplying every term inside the bracket by the term on the outside and then collecting like terms with the aim of removing the set of brackets. Expanding brackets is often an important step in solving equations and is the opposite process to factorisation.

The formula for expanding a single bracket is $a(b+c) = ab + ac$

The formula for expanding a double bracket is $(a+b)(c+d) = a(c+d)+b(c+d)=ac + ad + bc + bd$

This last formula for the product is often referred to as the FOIL method: Multiply the First terms, Outside terms, Inside terms, Last terms. $a$ and $c$ are the first terms, $a$ and $d$ are the outside terms, $b$ and $c$ are the inside terms and $b$ and $d$ are the last terms.

Note: The process of removing brackets is the same as the distributive law.

#### Worked Examples

###### Example 1

Expand $2x(xy-3x^2)$.

###### Solution

Start by multiplying $2x$ by the first term inside the bracket, then by the second. $2x(xy-3x^2) = 2x \times xy-2x\times 3x^2=2x^2y - 6x^3$

###### Example 2

Expand $(3x-4)(8-2x)$.

###### Solution

Start by multiplying $3x$ by $8$ and $-2x$, then multiply $-4$ by $8$ and $-2x$. \begin{align} (3x-4)(8-2x)&= 3x(8-2x)-4(8-2x)\\ &=24x -6x^2 -32 +8x\\ &=32x -6x^2-32 \end{align}

#### Factorisation

Factorization involves writing an expression as a product of factors. It is the opposite process of expanding brackets. A good way of checking if you have factorised an expression correctly is therefore to expand the brackets.

###### Example 1

Factorise the following where possible:

a) $2x+4y$

b) $2x+4xy$

c) $-4a+19abc$

d) $16xyz+x+8xy$

###### Solution

a) Both of the terms in this expression have a common factor of $2$. We can therefore bring this factor outside brackets as follows: $2x+4xy=2(x+2xy)$

b) Both of the terms in this expression have a common factor of $2x$ so we can factorise it as: $2x+4xy=2x(1+2y)$

c) The terms in this expression have a common factor of $a$ so we have: $-4a+19abc=a(19bc-4)$

c) All of the terms in this expression have a common factor of $x$ so we have: $16xyz+x+8xy=x(16yz+1+8y)$

#### Video Examples

###### Example 1

Prof. Robin Johnson expands the expression $x(2x-1)(2-x)$.

###### Example 2

Prof. Robin Johson expands the expression $(x-y)(x+y)$ and talks about the difference of two squares, which comes into use when factorising.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

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