In this paper, an explicit solution to polynomial matrix right coprime factorization of input-state transfer function is obtained in terms of the Krylov matrix and the Pseudo-controllability indices of the pair of coefficient matrices.The proposed approach only needs to solve a series of line ar equations. Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated.These new solutions are simple, they possess better structural properties and are very convenient to use. An example shows the effect of the proposed results.