# Relationship between coefficients of a polynomials and its roots.

The roots of the equation $x^4 -3x^2 + 5x - 2 = 0$ are $\alpha$, $\beta$, $\gamma$ and $\delta$. $\alpha^n + \beta^n + \gamma^n + \delta^n$ is denoted by $S(n)$. Find values of $S(2)$ and $S(4)$ and of $S(3)$ and $S(5)$.

Hence, find the value of $\alpha^2 (\beta^3 + \gamma^3 + \delta^3) + \beta^2 ( \alpha^3 + \gamma^3 + \delta^3) + \gamma^2 (\alpha^3 + \beta^3 + \delta^3) + \delta^2 (\alpha^3 + \beta^3 + \gamma^3)$

This topic is a rather strange one, and I can't seem to locate the formulas I need, and I need them for my exam. If you do use any formula (except $\sum\alpha$, $\sum\alpha\beta$, $\sum\alpha\beta\gamma$ and $\alpha\beta\gamma\delta$), please state it. Thank you in advance.