It is already well known that polynomials have a great theoretical impor-tance, because they can approximate any continuous function over an interval, with a relatively small error. Nowadays, various interpolation algorithms ex-ist in the literature (see [1,2]), and they propose approximating the value of a function at a point in an interval, knowing a finite number of values in points (nodes) of that interval. However, we can also program the computing of coef-ficients of the interpolating polynomial (Newton or Lagrange), if we take into account the formulas they proposed and the Vi`ete’s relations. The intention of this ’approach’ is to have access, through the same program, not only to the approximate value of the function at a specified point, but also to its interpolat-ing polynomial. We have thus tried to construct two interpolation algorithms, which provide the possibility of obtaining both the interpolating polynomial and the approximate value of the function at the specified point. Starting from the formulas developed by Newton and Lagrange, and from the relatively simple idea of using the Vi`ete’s relations (see [3]) for obtaining the coefficients of inter-polating polynomials (Newton or Lagrange), two algorithms were obtained, and then implemented, and they allow an effective access to the coefficients of the interpolant polynomial of a function, and so to its approximate values at various points belonging to the studied interval. These algorithms also bring, in addi-tion to obtaining the approximate value at a point, the “visualization” of the interpolating polynomial by which this approximation was achieved. Because of practical (numerical) reasons, Newton’s polynomial interpolation is privileged to Lagrange interpolation, and since the approach (in terms of the algorithms we developed) is similar for both types of polynomials, we will present in de-tail the Newton interpolation, for Lagrange interpolation we will highlight only briefly the steps of the algorithm, so it can then be programmed using the de-sired programming environment. We chose to use the C language because it has been used in our courses corresponding to the first year students classes. The programs obtained are then run on a few examples (which we extracted from our own activities in the courses/laboratories classes), to test their functionality. The running tests performed confirm the functionality of these algorithms, as well as their usefulness.