How can i reconstruct a polynomial with access to its roots and not to some points evaluation of it? Is there any other element that can let me interpolate a polynomial other than the well known lagrange interpolation by which a polynomial of degree $d$ can be reconstructed from $d+1$ pairs: $x_i,f(x_i)$. If i have another kind of information which is related with the polynomial, ie: some roots can i reconstruct it?

If you know all the roots of a polynomial, then you can almost reconstruct it. If $a_1,\dots,a_n$ are the roots, then the polynomials with exactly those roots are of the form $c(x-a_1)(x-a_2)\cdots (x-a_n)$ for some constant $c$ (assuming we really know all roots, ie that we are not necessarily limiting ourselves to the real roots or something like that).