This article addresses the algorithmics of linear systems with polynomial coefficients in one variable, highlighting their similarities with the treatment of rational fractions. Emphasis is placed on the importance of efficiently utilizing fast matrix multiplication. We consider the linear system in the form 𝐴(𝑋)𝑌(𝑋) = 𝐵(𝑋), where 𝐴 is an 𝑛 × 𝑛 matrix of polynomials with a non-zero determinant and 𝐵 is a vector of polynomials. We establish clear notations to facilitate understanding of the concepts. The article also presents complexity estimates related to matrix multiplication and evaluationinterpolation. Key results include the computation of the series expansion of 𝐴 −1𝐵 and the reconstruction of the coefficients of the rational fraction vector 𝑌 using Padé approximants. Newton’s method is discussed for its efficiency in the case of polynomial matrices. Finally, a detailed analysis of resolution algorithms, including those of Storjohann, is provided, highlighting recent advances in computing the coefficients of rational solutions of linear systems. The algorithmics of linear systems with polynomial coefficients in one variable is very similar to that of rational fractions. Moreover, it is important to leverage fast matrix multiplication.