What Are the Parentheses Rules?

On this page, we can see all the rules for parentheses in one spot.

Rule

Parentheses

$\begin{array}{llll}\hfill \phantom{\rule{-0.17em}{0ex}}\left(a+b\right)\phantom{\rule{-0.17em}{0ex}}\left(c+d\right)& =a\cdot c+a\cdot d\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}+b\cdot c+b\cdot d\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill a+\phantom{\rule{-0.17em}{0ex}}\left(c+d\right)& =a+c+d\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill a-\phantom{\rule{-0.17em}{0ex}}\left(c+d\right)& =a-c-d\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill a\cdot \phantom{\rule{-0.17em}{0ex}}\left(b+c\right)& =a\cdot b+a\cdot c\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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

Note! Remember to write the terms in descending order! Then it will look like this:

 $2{x}^{3}-4{x}^{2}+5x-3$

 $2{x}^{3}-4{x}^{2}+5x-3\phantom{\rule{2em}{0ex}}$

Example 1

Evaluate $x+\phantom{\rule{-0.17em}{0ex}}\left(-2x+5\right)$

$\begin{array}{llll}\hfill x+\phantom{\rule{-0.17em}{0ex}}\left(-2x+5\right)& =x-2x+5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-x+5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 2

Evaluate $4-\phantom{\rule{-0.17em}{0ex}}\left(12-3{x}^{2}\right)$

$\begin{array}{llll}\hfill 4-\phantom{\rule{-0.17em}{0ex}}\left(12-3{x}^{2}\right)& =4-12+3{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =3{x}^{2}-8\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 3

Evaluate $-4{x}^{2}\phantom{\rule{-0.17em}{0ex}}\left(2-x\right)$

$\begin{array}{llll}\hfill -4{x}^{2}\phantom{\rule{-0.17em}{0ex}}\left(2-x\right)& =-8{x}^{2}+4{x}^{3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =4{x}^{3}-8{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 4

Evaluate $\phantom{\rule{-0.17em}{0ex}}\left(x+1\right)\phantom{\rule{-0.17em}{0ex}}\left(x-2\right)$

$\begin{array}{llll}\hfill & \phantom{=}\phantom{\rule{-0.17em}{0ex}}\left(x+1\right)\phantom{\rule{-0.17em}{0ex}}\left(x-2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={x}^{2}-2x+x-2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={x}^{2}-x-2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{lll}\hfill \phantom{\rule{-0.17em}{0ex}}\left(x+1\right)\phantom{\rule{-0.17em}{0ex}}\left(x-2\right)={x}^{2}-2x+x-2={x}^{2}-x-2& \phantom{\rule{2em}{0ex}}& \hfill \end{array}$

Example 5

Evaluate $\phantom{\rule{-0.17em}{0ex}}\left(3{x}^{2}+y\right)\phantom{\rule{-0.17em}{0ex}}\left(2-x\right)$

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

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

Example 6

Evaluate $\phantom{\rule{-0.17em}{0ex}}\left(2x+3\right)\phantom{\rule{-0.17em}{0ex}}\left(4x-5\right)$

$\begin{array}{llll}\hfill & \phantom{=}\phantom{\rule{-0.17em}{0ex}}\left(2x+3\right)\phantom{\rule{-0.17em}{0ex}}\left(4x-5\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2x\cdot 4x+2x\cdot \phantom{\rule{-0.17em}{0ex}}\left(-5\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+3\cdot 4x+3\cdot \phantom{\rule{-0.17em}{0ex}}\left(-5\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8{x}^{2}+\phantom{\rule{-0.17em}{0ex}}\left(-10x\right)+12x-15\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8{x}^{2}+2x-15\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \phantom{\rule{-0.17em}{0ex}}\left(2x+3\right)\phantom{\rule{-0.17em}{0ex}}\left(4x-5\right)& =2x\cdot 4x+2x\cdot \phantom{\rule{-0.17em}{0ex}}\left(-5\right)+3\cdot 4x+3\cdot \phantom{\rule{-0.17em}{0ex}}\left(-5\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8{x}^{2}+\phantom{\rule{-0.17em}{0ex}}\left(-10x\right)+12x-15\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8{x}^{2}+2x-15\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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