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The problem is asking to solve \( x^3 - x - 1 = 0 \) on the interval \([1, 2]\) using a **fixed-point iteration method**, with the initial guess \( p_0 = 1 \), and find the solution to an accuracy of \( 10^{-2} \).

### Steps to Solve:

1. **Rearrange the equation into a fixed-point form**:  
   The fixed-point iteration method works by rearranging the equation in the form \( x = g(x) \). There are several possible ways to do this for the given equation. One simple rearrangement is:
   \[
   x = \sqrt[3]{x + 1}
   \]
   So, we define the function:
   \[
   g(x) = \sqrt[3]{x + 1}
   \]

2. **Start with the initial guess \( p_0 = 1 \)** and apply the iteration formula:
   \[
   p_{n+1} = g(p_n)
   \]

3. **Iterate until the difference between successive approximations \( |p_{n+1} - p_n| \) is less than \( 10^{-2} \)**.

Let's now compute this iteratively.After 4 iterations using the fixed-point iteration method, the solution is approximately \( p \approx 1.3243 \), with an error of about \( 0.0019 \), which is within the desired tolerance of \( 10^{-2} \).

Would you like further clarification or details on any part of the process?

Here are 5 related questions that build on this concept:

1. How do we choose the rearrangement \( g(x) \) for fixed-point iteration?
2. What are the conditions for convergence of the fixed-point iteration method?
3. Could we use another form of \( g(x) \) and still get convergence?
4. What happens if we change the initial guess \( p_0 \) in the fixed-point iteration method?
5. How does the fixed-point iteration compare to Newton's method in solving nonlinear equations?

**Tip**: When using the fixed-point method, always check that the function \( g(x) \) satisfies the conditions for convergence (typically \( |g'(x)| < 1 \) near the root).

Use a fixed-point iteration method to determine a solution accurate to within 10^{-2} for x^3 - x - 1 = 0 on [1, 2]. Use p_0 = 1.

Learn how to solve the equation x^3 - x - 1 = 0 using the fixed-point iteration method. This step-by-step solution explains how to rearrange the equation, apply iteration, and achieve an accuracy within 10^{-2}. Ideal for college-level students studying numerical methods.

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