We define a sequence of new higher-order polynomial invariants, delta_n(psi), for a given 3-manifold X and a class psi in H_2(X,bd(X)). These are especially important inasmuch as they give lower bounds for the Thurston norm of psi and give obstructions to X fibering over the circle. Using results of P. Kronheimer, T. Mrowka and S. Vidussi, we show that these also give obstructions to a 4-manifold of the form X x S^1 admitting a symplectic structure. In a recent paper, Curtis McMullen generalizes the well known result that for a knot K, the degree of the Alexander polynomial of K is a lower bound for twice the genus of K. He defines the Alexander norm of a class psi in H_2(X,bd X) for a given 3-manifold X and shows that if the first betti number of X is at least two then the Alexander norm is a lower bound for the Thurston norm. Our higher-order invariants delta_n agree with the Alexander norm when n=0 and can give "better" estimates for the Thurston norm than does the Alexander norm. We exhibit 3-manifolds whose Alexander norm is trivial but whose delta_n(psi) are strictly increasing and can be made arbitrarily large.