Some algebraic criteria for synchronization with error bound of forced oscillators with different external forces are derived by Lyapunov direct method and Sylvester's criterion. Using algebraic inequalities, the synchronization error bound can be estimated. By controlling the estimated synchronization error bound, the real synchronization error bound can be controlled to be smaller than a prescribed precision of the synchronization. The Duffing-Van der Pol forced oscillator used as a numerical example verifies the effectiveness of the method.