  Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m})$.
By $\mathcal{S}_j(M)$, we denote the $j$th symmetric power of $M$ ($j$th graded
component of the symmetric algebra $\mathcal{S}_R(M)$). The purpose of this
paper is to investigate the minimal free resolutions $\mathcal{S}_j(M)$ as
$R$-module for each $j\geq 2$ and determine the Betti numbers of
$\mathcal{S}_j(M)$ in terms of the Betti numbers of $M$. This has some
applications, for example for linear type ideals $I$, we obtain formulas of the
Betti numbers $I^j$ in terms of the Betti numbers of $I$. In addition, we
establish upper and lower bounds of Betti numbers of $\mathcal{S}_j(M)$ in
terms of Betti numbers of $M$. In particular, obtain some applications of the
famous Buchsbaum-Eisenbud-Horrocks conjecture.
