Polynomial evaluation at multiple points

Not sure if this is the right subforum for this, but anyways...

Suppose I have a large degree polynomial which I wanted to evaluate at multiple points. Is there a way one can do this faster than just directly evaluating the polynomial at each individual point?

For example suppose I wanted to evaluate the polynomial $\displaystyle x^8 + x^4 + 1$ at the points $\displaystyle x = 2,2^2,2^3,2^4,...$. Is there a faster way to do this than just evaluating each individual point?