Coefficients of a polynomial in terms of its roots
Assume you have a polynomial $p(x)$ of degree $n$ with roots $a_1, a_2, \dots, a_n$.
Let $S$ be the set of all roots. Let $S_i$ be the set of all subsets of $S$ with exactly $i$ elements. For example, if $S = \{ 1, 2, 3 \}$ then $S_2 = \{ \{1, 2\}, \{1, 3\}, \{2, 3\} \}$.
Let $\alpha_i = (-1)^i\Sigma\{ \Pi(y) \mid y \in S_i \}$.
Then
\[ p(x) = x^n + \alpha_1x^{n-1} + \alpha_2x^{n-2} + \cdots + \alpha_{n-1}x + \alpha_n \]