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.