This paper defines a new class of hybrid systems called piecewise polynomial (PWP) systems in strict form and develops a backstepping controller synthesis methodology for these systems. The main contribution of the paper is to formulate controller design for a large class of PWP systems as a convex feasibility problem. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares (SOS) Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a piecewise polynomial Lyapunov function for PWP systems with continuous vector fields. After constructing a (piecewise) polynomial Lyapunov function, controller synthesis for a PWP system can be formulated as an SOS program, which is a convex optimization problem and can be solved efficiently using available software. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to a numerical example.