$2\mathrm{.}13$ How much time is required to compute $f(x)={\displaystyle \underset{i=0}{\overset{N}{}}}{a}_i{x}^i$ :

a$.$ Using a simple routine to perform exponentiation?

b$.$ Using the routine in Section $2\mathrm{.}4\mathrm{.}4?$

$2\mathrm{.}14$ Consider the following algorithm $($known as Horner's rule$)$ to evaluate $f(x)=$

$)=n$;$i(0$;$-m$

a$.$ Show how the steps are performed by this algorithm for $x=3,f(x)=4x^4+8x^3+x+2$

b$.$ Explain why this algorithm works.

c$.$ What is the running time of this algorithm?