|Abstrakt:||This paper describes the properties of the Bernstein polynomial basis, concerning the computation of the intersection points of parametric and algebraic curves. The basic properties and connections between the power basis, the Bernstein basis and the scaled Bernstein basis are presented, and it is shown that the Bernstein basis has the best numerical stability in comparison to other bases. The resultant expression is defined, and the construction and properties of several formulations of resultant matrices are described. In addition to the widely known Sylvester's matrix, a resultant matrix of lower order -- the companion matrix resultant -- for two polynomials is obtained directly in the Bernstein basis, in terms of the companion matrix of one of the polynomials. The transformation of these resultants from the power basis to the Bernstein basis is also considered. The presented results of the elimination theory are then applied to compute the intersection of two polynomial curves from an algebraic approach, and a numerical solution for this problem is also included.