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).