Variation on variation on Euclid´s algorithm

In a paper entitled "Variation on Euclid\´s Algorithm for Plynomials", Calvez et al. has shown that the extended Euclid\´s algorithm can be partially obtained by the nonextended one; in fact, it can obtain only two of the three unknowns of the Bezout\´s theorem. This letter goes further and shows that all polynomials given by the extended Euclid\´s algorithm and all the intermediate values can be obtained directly by the nonextended Euclid\´s algorithm. Consequently, only remainder computations are used. Avoiding multiplications and divisions of polynomials decreases the computational complexity. This variation of Calvez et al. justifies the title of the present letter.