Suppose $P(x)$ is a polynomial of degree $n$ with roots $r_1, r_2, \ldots, r_n$ (not necessarily distinct). For $1\leq k\leq n$ , define $S_k$ by $$ S_k = \sum\limits_{1\leq\alpha_{1} < \alpha_{2} < \ldots\alpha_k\leq n} r_{\alpha_1}r_{\alpha_2}\ldots r_{\alpha_k $$ For example, $$ S_1 = r_1 + r_2 + r_3 + \ldots + r_ $$ $$ S_2 = r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + \ldots + r_{n-1}r_{n $$ Then writing $P(x)$ as $$ P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots a_{1}x + a_{0} $$ we find that $$ S_i = (-1)^{i}\frac{a_{n-i}}{a_n $$

For example, if $P(x)$ is a polynomial of degree 1, then $P(x) = a_1x + a_0$ and clearly $r_1 = -\frac{a_0}{a_1}$ .

If $P(x)$ is a polynomial of degree 2, then $P(x) = a_2x^2 + a_1x + a_0$ and $r_1 + r_2 = -\frac{a_1}{a_2}$ and $r_1r_2 = \frac{a_0}{a_2}$ . Notice that both of these formulas can be determined from the quadratic formula.

More intrestingly, if $P(x) = a_3x^3 + a_2x^2 + a_1x + a_0$ , then $r_1 + r_2 + r_3 = -\frac{a_2}{a_3}$ , $r_1r_2 + r_2r_3 + r_3r_1 = \frac{a_1}{a_3}$ , and $r_1r_2r_3 = -\frac{a_0}{a_3}$ .