Let $p$ and $q$ be fixed and $z_{1}, z_{2}$ and $z_{3}$ the three roots of the polynomial $X^{3}+p X+q$. Calculate $\frac{1}{z_{1}^{2}}+\frac{1}{z_{2}^{2}}+\frac{1}{z_{3}^{2}}$ in terms of $p$ and $q$.
We immediately exploit the relations between coefficients and roots:

$$
\left\{\begin{array}{cc}
\sigma_{1}=z_{1}+z_{2}+z_{3} & =0 \\
\sigma_{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1} & =p \\
\sigma_{3}=z_{1} z_{2} z_{3} & =-q
\end{array}\right.
$$

We then calculate

$$
\frac{1}{z_{1}^{2}}+\frac{1}{z_{2}^{2}}+\frac{1}{z_{3}^{2}}=\frac{z_{1}^{2} z_{2}^{2}+z_{2}^{2} z_{3}^{2}+z_{3}^{2} z_{1}^{2}}{\left(z_{1} z_{2} z_{3}\right)^{2}}=\frac{\left(z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\right)^{2}-z_{1} z_{2} z_{3}\left(z_{1}+z_{2}+z_{3}\right)}{\left(z_{1} z_{2} z_{3}\right)^{2}}=\frac{\sigma_{2}^{2}-\sigma_{3} \sigma_{1}}{\sigma_{3}^{2}}=\frac{p^{2}}{q^{2}}
$$