I know that in order to factor a one dimensional polynomial one can find the roots with some method, for instance a numerical newton method. Then one can systematically divide with $(variable-root)$ for each root found and then be done. Is there any analogous way to do this for multivariate polynomials? Does there exist any "unique" or "natural" factorization for those? It is obvious we can do this in the case our polynomial is separable i.e. $$P(x_1,x_2,\cdots,x_n) = P_1(x_1)P_2(x_2) \cdots P_n(x_n)$$ because then we could just factor each $P_k(x_k)$ separately.
But what about the general case?