Horner's Rule is an algorithm to efficiently evaluate a polynomial

$f(x)={\displaystyle \underset{i=0}{\overset{N}{}}}{a}_i{x}^i=a_N{x}^N+a_{N-1}{x}^{N-1}+cdots{a}_{1}x+a_0$

Here's the algorithm where the polynomial coefficients $(a_0,a_1,$dots,$a_N)$ are all stored in

an array or vector container $a$ :

poly $=0$;

for $($int i $=$ N; i$>=0$; i $--)$

poly $=$ x$*$ poly $+$a$[$i$]$

$($a$)$ Walk through the steps and show how the algorithm is used to calculate $f(2)$ for

$f(x)=3x^4-2x^3+x^2-7($so $x=2$ in this case$).$

$($b$)$ Explain why this algorithm works.

$($c$)$ Find the running time and explain why it is what it is$.$