Math Brackets Expanding brackets

# Expanding brackets

If you expand brackets you use the distributive law.

## A bracket is multiplied with a factor

When a bracket is expanded, each term within the bracket is multiplied by the factor outside the bracket. Here, + and - are set according to the sign rules. The following applies:

$\color{red}{a}\cdot(b+c)=\color{red}{a}\cdot b+\color{red}{a}\cdot c$

### Examples

• $5 \cdot (x+y)=5\cdot x+5\cdot y$

• $-2\cdot (5x-2)$ $=-2\cdot 5x+(-2)\cdot (-2)$ $=-10x+4$

• $(1-y)\cdot x=x\cdot 1+x\cdot (-y)$ $=x-xy$

## A bracket is multiplied with a bracket

If you expand two brackets, each member of one bracket is multiplied by each member of the other bracket. Here, + and - are set according to the sign rules. The following applies:

$(\color{red}{a}+\color{blue}{b})\cdot(c+d)=\color{red}{a}c+\color{red}{a}d+\color{blue}{b}c+\color{blue}{b}d$

### Examples

• $(4a+3b)\cdot(5x−8y)$ $=20ax−32ay+15bx−24by$

• $(2a-4b)\cdot(-7a-2b+9)$
$=-14a^2-4ab+18a+28ab$ $+8b^2-36b$
$=-14a^2+24ab+18a+8b^2-36b$