# How do you factor (r+s)(s+t)-(r+s)(s-t)+(r+s)(s+t)?

Mar 3, 2018

$\left(r + s\right) \times \left\{\left(s + t\right) - \left(s - t\right) + \left(s + t\right)\right\}$

#### Explanation:

To factor this expression, look for a factor that all the terms have in common.

Then you can factor that one out from all the terms.

In this case, all three terms have a factor in common, namely $\left(r + s\right)$.

Factor out $\left(r + s\right)$ from each term

$\left(r + s\right) \left\{\textcolor{w h i t e}{\frac{2}{2}} \left(s + t\right) - \left(s - t\right) + \left(s + t\right) \textcolor{w h i t e}{\frac{2}{2}}\right\}$ $\leftarrow$ answer

$\textcolor{w h i t e}{m m m m m}$ ―――――――――

• Be sure to keep the parentheses for the expression you factored out

It's an error to write it like this (without the beginning parentheses)

r + s  {(s + t) - (s - t) +  (s + t)}
This means that only the $s$ is distributed to all the terms in the brackets instead of distributing the entire $\left(r + s\right)$

• Be sure to enclose the expression in brackets

It's an error to write it like this (without the brackets)

$\left(r + s\right) \times \left(s + t\right) - \left(s - t\right) + \left(s + t\right)$
This doesn't show that $\left(r + s\right)$ is to be distributed to all the terms. It makes it look like $\left(r + s\right)$ should multiply only $\left(s + t\right)$

$\textcolor{w h i t e}{m m m m m}$ ―――――――――

Check
To check factoring, see if distributing the factor brings back the original expression

$\left(r + s\right) \times \left\{\left(s + t\right) - \left(s - t\right) + \left(s + t\right)\right\}$

Distribute $\left(r + s\right)$ to each of the terms

$\left(r + s\right) \left(s + t\right) - \left(r + s\right) \left(s - t\right) + \left(r + s\right) \left(s + t\right)$

$C h e c k$

Mar 3, 2018

$\left(r + s\right) \left(s + 3 t\right)$

#### Explanation:

$\text{take out the "color(blue)"common factor } \left(r + s\right)$

$\Rightarrow \left(r + s\right) \left[\left(s + t\right) - \left(s - t\right) + \left(s + t\right)\right]$

$\text{simplifying the terms in the bracket gives}$

$= \left(r + s\right) \left(\cancel{s} + t \cancel{- s} + t + s + t\right)$

$= \left(r + s\right) \left(s + 3 t\right)$