This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $P_n^{(\alpha ,\beta )}(x)$ and introduces an explicit formula based on the zeros of $P_{n-1}^{(\alpha ,\beta )}(x)$. Several illustrative examples are included.