In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan-Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan-Lusztig, the inverse Kazhdan-Lusztig and the Z-polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are nondegenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan-Lusztig, inverse Kazhdan-Lusztig and Z-polynomial of all sparse paving matroids.Conjecture 1.1 ([13, Conjecture 3.2]). The Kazhdan-Lusztig polynomial of a matroid is real-rooted.