In this article, we show how to perform a dynamical control of Newton's method for the computation of multiple roots of polynomials. Using Discrete Stochastic Arithmetic, root approximations are computed until the difference between two successive approximations is numerical noise. With such a stopping criterion, the optimal number of iterations in Newton's method are performed. Moreover it is possible to estimate in the result obtained which digits are in common with the exact root. Two strategies to estimate the multiplicity of polynomials roots are compared: one requires root approximations computed at different precisions and the other three successive iterates of Newton's method. We show that using such a strategy and then the modified Newton's method, multiple roots can be computed with a requested accuracy.