Consider we have $n$ pairs of $(x_i,y_i)$. We all know that given the $n$ pairs we can interpolate a polynomial of degree at most $n-1$.
Also, it is clear if we want to find roots of a interpolating polynomial, (given the $n$ pairs); we need to:
1- interpolate the polynomial.
2- find the roots of it.
Question: given the $n$ pairs, is there any way (or shortcut) to find roots of interpolating polynomial, without interpolating the polynomial first?
Note: I am after a faster way of finding the interpolating polynomial's roots than going through the above two steps. However, I DO NOT know whether possible to do it.
The knowledge of $(x_i,y_i)$ directly gives you only a vague idea where some of the zeros are: if two adjacent $y$-values have opposite signs, there is a zero between them. Otherwise, there is no shortcut to get from the points to zeros.
Indeed, consider an arbitrary polynomial $p$ of $n$th degree. You can pick any $n+1$ points, evaluate $p$ at them, and thus get the point set $(x_i,y_i)$ for which $p$ is the interpolating polynomial. If there was a shortcut to the zeroes of $p$ from here, we'd have a new easy way to find the roots of any polynomial. But finding roots is hard; it's far more difficult then constructing the interpolating polynomial.