# How do you find the product (f+g)(f-g)(f+g)?

May 31, 2017

${f}^{3} + {f}^{2} g - f {g}^{2} - {g}^{3}$

#### Explanation:

Use expansion to simplify this expression.

Expansion is conventionally done left to right.

Multiply the terms in order as shown with colours.

$\left(\textcolor{red}{f} + g\right) \left(\textcolor{red}{f} - g\right) \left(f + g\right)$

$\left(\textcolor{red}{f} + g\right) \left(f - \textcolor{red}{g}\right) \left(f + g\right)$

$\left(f + \textcolor{red}{g}\right) \left(\textcolor{red}{f} - g\right) \left(f + g\right)$

$\left(f + \textcolor{red}{g}\right) \left(f - \textcolor{red}{g}\right) \left(f + g\right)$

Therefore, $\left(f + g\right) \left(f - g\right) \left(f + g\right)$

$= \left({f}^{2} - f g + f g - {g}^{2}\right) \left(f + g\right)$

$= \left({f}^{2} - {g}^{2}\right) \left(f + g\right)$

Next, follow similar steps as above.

$\left(\textcolor{red}{{f}^{2}} - {g}^{2}\right) \left(\textcolor{red}{f} + g\right)$

$\left(\textcolor{red}{{f}^{2}} - {g}^{2}\right) \left(f + \textcolor{red}{g}\right)$

Repeat for $- {g}^{2}$.

$= \left({f}^{2} - {g}^{2}\right) \left(f + g\right)$

$\textcolor{b l u e}{= {f}^{3} + {f}^{2} g - {g}^{2} f - {g}^{3}}$