This paper proposes the synchronization control method for two different unified chaotic systems with unknown mismatched parameters using sum of squares method. Previously, feedback-linearizing and stabilization terms were used in the controller for the synchronization problem. However, they used just a constant matrix as a stabilization control gain, whose performance is shown to be valid only for a linear model. Thus, we propose the novel control method for the synchronization of the two different unified chaotic systems with unknown mismatched parameters via sum of squares method. We design the stabilization control input which is of the polynomial form by sum of squares method and also the adaptive law for the estimation of the unknown mismatched parameter between the master and slave systems. Since we can use the polynomial control input which is dependent on the system states as the stabilization controller, the proposed method can have better performance than the previous methods. Numerical simulations for both uni-directional and bi-directional chaotic systems show the validity and advantage of the proposed method.