The researches for Bezier and polynomial curves and surfaces, and their applications to curve- fitting have been reported in many papers. However the relation between control points and polynomial coefficients, because of the complexity of computation, have rarely been studied. In this paper, we propose a new method to transform the Bezier to the polynomial representation and vice-versa. An equation is given, for generating an (m + 1) polygonal Bezier control points to approximate an (n + 1) ones. This method, unlike previous works, is more transparent because it is given in form of one equation. With this method, the curve goes through the two endpoints of the polygonal, and we do not need to perform any transformation such as Chebyshev polynomial in order to obtain good approximation. A criterion of reduction is given in order to known if a polygonal Bezier is reducible without error or not. An error estimation is also given only in terms of control points. All equations are given explicitly and in matrix form, for Bezier curves. Finally, we discuss some applications of this method to curve-fitting, order increasing and decreasing, and also its extension to rational Bezier and polynomial.