Consider the polynomial equation

$a_0{x}^n{a}_1{x}^{n-1}$ cdots $a_{n-1}x{a}_n=0,a_{0}0$

with $($complex$)$ roots $r_1,r_2,$dots,$r_n,$ so that

$a_0{x}^n{a}_1{x}^{n-1}$ cdots $a_{n-1}x{a}_n$

$=a_0(x-r_1)(x-r_2)cdots(x-r_n)$

By multiplying the linear factors on the right and comparing the resulting coefficients, prove that

$r_1{r}_2{r}_3$ cdots $r_n=-\frac{a_1}{a_0}$

$r_1{r}_2{r}_1{r}_3{r}_2{r}_3$ cdots $r_{n-1}{r}_n=\frac{a_2}{a_0}$

vdots

$r_1{r}_2{r}_{3}cdots{r}_n=(-1{)}^n\frac{a_n}{a_0}$

where the $k$ th equation asserts that the sum of all possible products of the roots, taken $k$ at a time, equals $(-1{)}^{\frac{a}{a}}\frac{a_t}{a_0}.$