I am wondering what methods exist for effectively evaluating polynomials (manually or in the head) in a quick, efficient fashion.
For example, one of my favorite methods is the "nested form of a polynomial." Consider the following:
- $P(x) = 3x^4-5x^3+x^2-3x+5$
- $Q(x) = (((3x-5)x+1)x-3)x+5$
When $Q$ is expanded, it is clear that $P = Q$. When trying to evaluate, say, $P(2)$ and $Q(2)$, it becomes obvious that $Q(2)$ is much easier to evaluate.
Similarly, consider two more polynomial expressions:
- $R(x) = x^5-2x^4+3x^3+3x+4$
- $S(x) = (((x-2)x+3)x^2+3)x+4$
Again, it is easy to see that $R=S$ when $S$ is expanded. It is also clear that trying to evaluate $S(3)$ in your head is much easier than trying to evaluate $R(3)$.
Calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division. It is also very easy, as the above examples illustrate, to construct the nested form of a polynomial.
Consideration: What other methods or "tricks" exist that allow one to quickly evaluate polynomials?