The exact representation of symmetric polynomials on Banach spaces with symmetric basis and also on separable rearrangement-invariant function spaces over [0, 1] and [0, ∞) is given. As a consequence of this representation it is obtained that, among these spaces, [lscr ]2n, L2n[0, 1], L2n[0, ∞) and L2n[0, ∞)∩L2m[0, ∞) where n, m are both integers are the only spaces that admit separating polynomials.