## Vieta's Formulas

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*}

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