Vieta's theorem states that given a polynomial
$$ a_nx^n + \cdots + a_1x+a_0$$
the quantities
$$\begin{align*}s_1&=r_1+r_2+\cdots\\
s_2&=r_1 r_2 +r_1 r_3 + \cdots \end{align*}$$
etc., where $r_1,\dots, r_n$ are the roots of the given polynomial, are given by
$$s_i = (-1)^i \frac{a_{n-i}}{a_n} .$$
So my question is: can we use this to find all the roots of a polynomial?
Best Answer
Quick answer: you're not going to find the roots in any quicker way with this method. Remember that in general, for polynomials of degree 5 or more, you cannot find explicit formulas for the roots. You simply cannot. With this method or another.
Now, what is Vieta's theorem? It is in fact just expanding the product $$ a_n \prod_{i=1}^n (x - r_i) = a_nx^n + \cdots + a_1x+a_0$$