We study rational generating functions of sequences {an } n≥0 that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences {arn } n≥0. We prove that if the numerator polynomial for {an } n≥0 is of degree s and its coefficients satisfy a set of natural linear inequalities, then the symmetric decomposition of the numerator for {arn } n≥0 is real-rooted whenever r = max{s,d+1-s}.Moreover, if the numerator polynomial for {an } n≥0 is symmetric, then we show that the symmetric decomposition for {arn } n≥0 is interlacing.We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the h*-polynomial of every dilation of a d-dimensional lattice polytope of degree s has a real-rooted symmetric decomposition whenever the dilation factor r satisfies r = max{s,d + 1 - s}. Moreover, if the polytope is Gorenstein, then this decomposition is interlacing.