For a polynomial of the form
\begin{align*} f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 \end{align*}
with roots \(r_1,r_2,r_3,...r_n \), Vieta's formulas state that:
\begin{align*} r_1+r_2+r_3+...+r_n=-\frac{a_{n-1}}{a_n} \\ r_1r_2 +r_1r_3+...+r_{n-1}r_n=\frac{a_{n-2}}{a_n} \\ r_1r_2r_3+r_1r_2r_4+...+r_{n-2}r_{n-1}r_n=-\frac{a_{n-3}}{a_n} \\ \vdots \\ r_1r_2r_3...r_n=(-1)^n\frac{a_0}{a_n} \\ \end{align*}
Generalization: When the \( n \) roots are taken in groups of \( k \) (i.e. \(r_1+r_2+...+r_n \) is taken in groups of \(1\) and \(r_1r_2...r_n \) is taken in groups of \( n \)), this is equivalent to
\begin{align*} (-1)^k\frac{a_{n-k}}{a_n} \end{align*}