Somehow I doubt there are useful methods, but maybe I have a blind spot. Assume with have a complex polynomial p with degree n and complex numbers α and β.
And we know p(α) and p(β), is there a quick way to compute p(α+β)? Or are there some particular polynomials p where there is such a method.
Edit: Doing something with the coefficients is allowed. We need to produce p(α), p(β) and p(α+β) at the same time. Everything that is faster than evaluating the polynomial 3-times is quicker.