Polynomial chaos theory provides a computationally efficient tool for control system design subject to probabilistic parametric uncertainties. In most existing methods, Galerkin projection is used to approximate the original stochastic system by a deterministic projected system of higher dimensions, so that control synthesis can be performed within the projected space. These methods have two main limitations: (i) a non-convex optimization problem is obtained for control synthesis; and (ii) due to approximation errors, stability derived for the projected approximation may not be automatically achieved by the original system. In this article, a new polynomial chaos based approach is proposed for stochastic discrete-time linear quadratic regulation. Instead of approximating the original system with Galerkin projection, a guaranteed cost problem is formulated and then approximated by using polynomial chaos. A tuning parameter is introduced to explicitly account for the approximation errors. A semidefinite program is then derived by exploiting orthogonality of the polynomial bases, in contrast to the non-convex optimization obtained by the Galerkin projection based methods. In particular, the general nonlinear parametric dependence can be effectively addressed by using its piecewise polynomial approximation. A numerical example illustrates the efficacy of our proposed approach.