Let $p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+$cdots$+a_0.$

Task:

Describe an algorithm on "paper" using pseudocode, that you can employ the Horner's algorithm

idea to efficiently calculate the Taylor polynomial $($coefficients$)$ of degree $mn$ of the given

polynomial $p(x)$ centered around any point $x_{0}0.$ Specifically, you need to determine the

coefficients $c_k=\frac{p^{(k)}(z_0)}{k!}$ for $k=0,1,$dots,$m.($you may choose to follow the hint below$)$

Briefly explain why this algorithm will also work if $x_0=1\mathrm{.}73x_{0}p(x_0)p(x_0)p(x)(x-x_0)(x-x_0)q(x)(x-x_0)q(x)=p(x)-p(x_0)q(x_0)m.$

Write the Matlab code $to$ efficiently implement this algorithm

Choose the polynomials from problem $3$ and $4$ and calculate the Taylor polynomial $of$ the same

degree centered $at{x}_0=1\mathrm{.}73$ using your code. Verify your code $by$ comparing the values $of$ the

original polynomial and its Taylor polynomial $at$ several points.

Test also your code $by$ computing the values $at$ those same points $as$ the previous bullet, $of$ thess

computed one$-$degree$-$less Taylor polynomials, around the same $x_0.$

Hint:

Start $by$ evaluating $p(x_0).$ What does $it$ give you?

Conceptually subtract $p(x_0)$ from $p(x)to$ form a new polynomial, and recognize that $it$ has $(x-x_0)$

$as$ a factor. Use $an$ idea similar $to$ Horner's method $to$ factor out $(x-x_0)$ iteratively, forming $a$

polynomial $q(x)$ such that $(x-x_0)q(x)=p(x)-p(x_0).$

Efficiently evaluate just the $q(x_0)$ what does $it$ give you now? Repeat this process.