It’s not like evaluating polynomial is something that comes up that frequently in the average programmer’s day. In fact, almost never, but it brings up a number of interesting questions, such as, how do we evaluate it very efficiently and how much of a simplification in the computation is actually a simplification?
The generic polynomial has the form
If we evaluate this naïvely, we end up doing multiply and additions. Even using the fast exponentiation algorithm, we still use multiplies. But we can, very easily, get down to .