Q$.$ Consider the following polynomial with a real variable, $x$;

$p(x)=3x^3-5x^2-16x+12$

The polynomial and its root are shown in the figure below.

a$.(15$ points$)$ Write a Python function for $p(x)$ that takes $x$ as a formal

parameter and returns $p(x).$

b$.(15$ points$)$ Write a Python function for the derivative of $p(x)$ that takes $x$ as

a formal parameter and returns the derivative of $p(x).$ Take the derivative of

$p(x)$ analytically with respect to $x$ before writing the function.

c$.(45$ points$)$ Write a Python function for root finding using the Newton$-$

Raphson method that takes an initial guess, $x_0,$ as a formal parameter, and

returns its root, $x_r,$ of $p(x)$ such that $p(x_r)0.$ At each iteration, the function is

expected to print the iteration number, the new guess and the absolute

value of $p(x)$ at the guess. Iterations should stop when the absolute value

of $p(x)$ at a new guess is less than ${10}^{-6}.$ The function will return that guess

as $x_r.$

d$.(25$ points$)$ Write a main program that will get the initial guess, $x_0,$ from a

user using the input statement, and call the functions developed earlier as

needed to find the root of the polynomial. It should call the function

developed for question $1.$b to find the root. The program is supposed to print

the root right after locating it$.$