How to Expand Parentheses with Variables

When you’re working with both variables and parentheses, you need to follow a particular set of rules. These rules allow you to remove the parentheses to clean up your expression. In school, you’ll spend a good amount of time working with expressions like this, so make sure to pay close attention.

Rule

NumbersandSignsMultipliedbyParentheses

1.
A $+$ in front of a parenthesis changes nothing inside the parentheses.
2.
A $-$ in front of a parenthesis changes all the signs inside the parentheses, so a $+$ becomes a $-$ and a $-$ becomes a $+$.
3.
A positive factor multiplied by a parenthesis leaves the signs unchanged, but the terms inside the parentheses are multiplied by the factor outside.
4.
A negative factor multiplied by a parenthesis changes all the signs inside the parentheses, and the terms inside the parentheses are multiplied by the factor outside.

Here are a couple of examples illustrating the four rules above:

Example 1

Rule1

Expand the parentheses:

 $+\left(x+1\right)=x+1$

Expand the parentheses:

 $+\left(x-1\right)=x-1$

Expand the parentheses:

 $+\left(-x-1\right)=-x-1$

Expand the parentheses:

 $+\left(-x+1\right)=-x+1$

Example 2

Rule2

Expand the parentheses:

$\begin{array}{llll}\hfill -\left(x+1\right)& =-+x-+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-x-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -\left(x+1\right)& =-+x-+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-x-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill -\left(x-1\right)& =-+x--1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -\left(x-1\right)& =-+x--1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill -\left(-x-1\right)& =--x--1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -\left(-x-1\right)& =--x--1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill -\left(-x+1\right)& =--x-+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -\left(-x+1\right)& =--x-+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 3

Rule3

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(a+3\right)& =+2a\cdot \left(+a\right)\phantom{l}+2a\cdot \left(+3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2{a}^{2}+6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(a+3\right)& =+2a\cdot \left(+a\right)\phantom{l}+2a\cdot \left(+3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2{a}^{2}+6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(a-3\right)& =+2a\cdot \left(+a\right)\phantom{l}+2a\cdot \left(-3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2{a}^{2}-6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(a-3\right)& =+2a\cdot \left(+a\right)\phantom{l}+2a\cdot \left(-3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2{a}^{2}-6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(-a-3\right)& =+2a\cdot \left(-a\right)\phantom{l}+2a\cdot \left(-3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-2{a}^{2}-6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(-a-3\right)& =+2a\cdot \left(-a\right)\phantom{l}+2a\cdot \left(-3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-2{a}^{2}-6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(-a+3\right)& =+2a\cdot \left(-a\right)\phantom{l}+2a\cdot \left(+3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-2{a}^{2}+6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill 2a\phantom{\rule{-0.17em}{0ex}}\left(-a+3\right)& =+2a\cdot \left(-a\right)\phantom{l}+2a\cdot \left(+3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-2{a}^{2}+6a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 4

Rule4

Expand the parentheses:

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(b+a\right)& =a\cdot \left(+b\right)\phantom{l}-a\cdot \left(+a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-ab-{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(b+a\right)& =-a\cdot \left(+b\right)\phantom{l}-a\cdot \left(+a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-ab-{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(b-a\right)& =-a\cdot \left(+b\right)\phantom{l}-a\cdot \left(-a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-ab+{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(b-a\right)& =-a\cdot \left(+b\right)\phantom{l}-a\cdot \left(-a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-ab+{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(-b-a\right)& =-a\cdot \left(-b\right)\phantom{l}-a\cdot \left(-a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =ab+{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(-b-a\right)& =-a\cdot \left(-b\right)\phantom{l}-a\cdot \left(-a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =ab+{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Expand the parentheses:

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(-b+a\right)& =-a\cdot \left(-b\right)\phantom{l}-a\cdot \left(+a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =ab-{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill -a\phantom{\rule{-0.17em}{0ex}}\left(-b-a\right)& =-a\cdot \left(-b\right)\phantom{l}-a\cdot \left(-a\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =ab-{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$