Summary: Routh’s method, which is widely known as a method for analyzing the stability of linear systems or, in a more general sense, as a method for determining the number of right-hand roots of a polynomial, has, since about the beginning of the 1960s, also been finding ever wider applications in the solution of other problems in systems theory. This article examines a number of problems in systems theory. They include analysis of the composition of the set of roots of a characteristic polynomial, determination of stability reserves and of frequency quality indices, analysis of the conditions of harmonic balance, absolute stability, the conditions under which rational functions are positive, determination of the boundary points of \(D\)-partitions, etc. These examples show the importance of certain typical problems in the algebra of polynomials for the theory of systems. The proposal is to generalize Routh’s problem as a set of three such problems: that of the Cauchy index of a rational function and the mutuality properties of the corresponding pair of polynomials, that of the composition of the set of roots of a polynomial in their distribution, in the complex plane of the roots, relative to the real axis, and that of the composition of the set of roots of a polynomial in their distribution relative to the imaginary axis. Complete solution of this problem on the basis of generalizations of Routh’s method and algorithm greatly widens the applications of the latter in the theory of systems and enables new approaches to be determined in constructing efficient tools for investigating a broad circle of problems in that theory.