This book considers structure preserving matrix methods for computations on Bernstein polynomials whose coefficients are corrupted by noise. The ill-posed operations of greatest common divisor computations and polynomial division are considered, and it is shown that structure preserving matrix methods yield excellent results. With respect to greatest common divisor computations, the most difficult part is the computation of its degree, and several methods for its determination are presented. These are based on the Sylvester resultant matrix, and it is shown that a modified form of the Sylvester resultant matrix yields the best results. The B´ezout resultant matrix is also considered, and it is shown that the results from it are inferior to those from the Sylvester resultant matrix.