In this paper we show that the test of Hurwitz property of a segment of polynomials (1 - lambda )p(0)(s) + lambdap(1)(s), where lambda is an element of [0,I], p(0)(s) and p(1)(s) are nth-degree polynomials of real coefficients, can be achieved via the approach of constructing a fraction-free Routh array and using Sturm's theorem. We also establish the connection between the proposed approach and the finite-step methods based on the resultant theory and the boundary crossing theorem. In a certain sense, the proposed approach provides an efficient numerical implementation of the later two methods and, therefore, by which the robust Hurwitz stability of convex combinations of polynomials can be checked in a definitely finite number of arithmetic operations without having to invoke any root-finding procedure.