$!!!!!$SOLVE THIS USING REQUIRED GUIDLINES ONLY BASIC THINGS SUCH AS LOOPS AND IF ELSE CONDITION CAN BE USED ITS A BASIC PROGRAM SO NO COMPLEX THING AND ONLY "IOSTREAM HEADER FILE" $($NO CHATGPT OR ANY AI PLS$)($C$++$ ONLY NO C PLS$)!!!!!!$

Exercise:

Q$1$: Halley's Method for Determination of roots of polynomial is

$x_{n+1}=x_n-\frac{2f(x_n)f^{\text{'}}(x_n)}{2f^{\text{'}}(x_n{)}^2-f(x_n)f^{\text{'}\text{'}}(x_n)}$

Write a program to find roots of the following polynomial using Halley's method, precision is $0\mathrm{.}0001$

$f(x)=a{x}^3+b{x}^2+cx+d$

Q$2$: Write an iterative algorithm to implement the following expansion $($precision upto $0\mathrm{.}0001)$

$sinx={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-$cdots, for all $x$PBL or OEL Statement:

Q: Write a program to find roots of the following polynomial using Halley's Method with precision

$0\mathrm{.}0001.$

$f(x)=a{x}^3+b{x}^2+cx+d$

Halley's method for determining roots of polynomial is$,$

$x_{n+1}=x_n-\frac{2f(x_n)f^{\text{'}}(x_n)}{2f^{\text{'}}(x_n{)}^2-f(x_n)f^{\text{'}\text{'}}(x_n)}$

Q: Write an iterative algorithm to implement the following expansion $($precision upto $0\mathrm{.}0001)$

$sin(x)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1}$

Deliverables:

$C++$ code containing the implementation of the above the algorithms.

TMethodology:

Follow methods provided in the lab manual for implementing above algorithms.

Guidelines:

Understand the concept of Halley's Method and how it can be used to find the roots of

a polynomial

Write a plan for the program, including the algorithm and data structures needed.

Write the code for the program, using appropriate data types and functions to

implement the Halley's Method algorithm.

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