Solutions:

1. Solving the Polynomial Equation $2x^4 + 3x^3 - 3x - 2 = 0$:

   • Using the Newton-Raphson method with an initial guess of $x_0 = 0.5$, the root found is approximately $x = 1.0$.
   • The method converged in 13 iterations.

2. Finding $\sqrt{5}$ using Newton-Raphson Method:

   • Using the Newton-Raphson method with an initial guess of $x_0 = 2.0$, the approximation for $\sqrt{5}$ is approximately $x = 2.23606797749979$.
   • The method converged in 3 iterations.

Detailed Steps for Newton-Raphson Method to Find $\sqrt{5}$:

1. Define the function and its derivative:

   • $f(x) = x^2 - 5$
   • $f'(x) = 2x$

2. Iterative Formula: $x_{n+1} = \frac{1}{2} \left( x_n + \frac{5}{x_n} \right)$

3. Iterations:

   • Start with $x_0 = 2.0$
   • $x_1 = \frac{1}{2} \left( 2.0 + \frac{5}{2.0} \right) = 2.25$
   • $x_2 = \frac{1}{2} \left( 2.25 + \frac{5}{2.25} \right) = 2.236111111111111$
   • $x_3 = \frac{1}{2} \left( 2.236111111111111 + \frac{5}{2.236111111111111} \right) = 2.236067977915804$

The value converges to $2.23606797749979$ after 3 iterations, which is the square root of 5.

Do you have any questions or would you like further details on any of the steps?

Here are some related questions that might interest you:

1. What are other numerical methods to solve polynomial equations?
2. How can we improve the convergence rate of the Newton-Raphson method?
3. What are the limitations of the Newton-Raphson method?
4. How do we choose an initial guess for the Newton-Raphson method?
5. Can the Newton-Raphson method fail to converge? If so, why?
6. What is the geometric interpretation of the Newton-Raphson method?
7. How do we extend the Newton-Raphson method to systems of nonlinear equations?
8. What are some real-world applications of solving polynomial equations?

Tip: When using the Newton-Raphson method, a good initial guess can significantly improve convergence speed and accuracy.