As it is well-known the Newton-Raphson method is closely connected with the Taylor polynomial. Using this connection the Ostrowski' fundamental existence theorem for Newton-Raphson method [3], [4] can be proved in an very natural way [6]. The S.B. Prešić's method [7] for simultaneuus determination of all roots of polynomial can be obtained using the interpolation formulae of Newton and Lagrange [5]. We use that fact in the convergence theorem which we prove in this paper. We note that the convergence conditions depend only on the intial points of roots, their distances and on the degree of polynomial.