# How do you expand (z+1/z)^2?

Nov 14, 2016

$= {z}^{2} + 2 + \frac{1}{z} ^ 2$

#### Explanation:

Do not be put off by the fraction.

${\left(z + \frac{1}{z}\right)}^{2} = \left(z + \frac{1}{z}\right) \left(z + \frac{1}{z}\right)$

Multiply the 2 brackets - the FOIL method if you use it.

$\left(\textcolor{b l u e}{z} + \textcolor{red}{\frac{1}{z}}\right) \left(z + \frac{1}{z}\right)$

$= \textcolor{b l u e}{{z}^{2} + 1} + \textcolor{red}{1 + \frac{1}{z} ^ 2} \text{ } \rightarrow \frac{\cancel{z}}{1} \times \frac{1}{\cancel{z}} = 1$

$= {z}^{2} + 2 + \frac{1}{z} ^ 2$