An analytically solvable polynomial optimal guidance law is proposed for the precision guidance problem considering multiple constraints on terminal miss distance, impact angle and acceleration. By setting the tangent value of look angle as a polynomial function of the relative distance between target and missile, the terminal multiple constraints are transformed into algebraic relations of polynomial coefficients. The optimization theory is introduced to obtain the optimal solution of polynomial coefficients under the energy optimal index weighted by the distance between target and missile. According to the motion relationship of missile and target, the tangent value of look angle is rewritten as the analytic expression of guidance instruction which satisfies the terminal impact angle and acceleration constraint. The guidance effect of guidance law is verified by simulation according to different weighted coefficients and different terminal impact angles, and the results are compared with the trajectory shaping guidance law. The simulation results show that the proposed guidance law is able to lead the missiles attack the target accurately at any desired impact angle, and the terminal acceleration instruction converges to 0 smoothly, which avoids the phenomenon of terminal instruction saturation. Compared with previous research on polynomial guidance, the guidance method avoids introducing linear approximation condition for small angles into the guidance model, which improves the accuracy of trajectory and guidance instruction design. At the same time it can constrain the look angle to avoid the singularity problem and the monotonicity problem of the independent variable that may occur in the system, and can attack in all directions at the terminal time.