Find the possible root of x³ - 2x²-5x+6=0 using the fixed point iteration method

To find a root of the equation \( x^3 - 2x^2 - 5x + 6 = 0 \) using the fixed point iteration method, we follow these steps:

1. **Rewrite the equation in the form \( x = g(x) \)**:
   We need to rearrange the equation \( x^3 - 2x^2 - 5x + 6 = 0 \) into a form suitable for fixed point iteration. One possible rearrangement is:
   \[
   x = \frac{x^3 - 2x^2 + 6}{5}
   \]
   Thus, we define our iteration function \( g(x) \) as:
   \[
   g(x) = \frac{x^3 - 2x^2 + 6}{5}
   \]

2. **Choose an initial guess**:
   An initial guess, \( x_0 \), is needed to start the iteration. For simplicity, let's start with \( x_0 = 1 \).

3. **Iteration formula**:
   The iteration formula is \( x_{n+1} = g(x_n) \).

4. **Iterate until convergence**:
   We perform iterations until the values converge to a desired level of accuracy.

Let's perform a few iterations:

### First iteration:
\[
x_1 = g(x_0) = g(1) = \frac{1^3 - 2 \cdot 1^2 + 6}{5} = \frac{1 - 2 + 6}{5} = \frac{5}{5} = 1
\]

### Second iteration:
\[
x_2 = g(x_1) = g(1) = \frac{1^3 - 2 \cdot 1^2 + 6}{5} = \frac{1 - 2 + 6}{5} = \frac{5}{5} = 1
\]

It appears that our initial guess \( x_0 = 1 \) has converged immediately, suggesting that \( x = 1 \) is a fixed point and hence a root of the equation.

### Verification:
Substitute \( x = 1 \) back into the original equation:
\[
1^3 - 2 \cdot 1^2 - 5 \cdot 1 + 6 = 1 - 2 - 5 + 6 = 0
\]
Thus, \( x = 1 \) is indeed a root.

### Conclusion:
The root found using the fixed point iteration method is \( x = 1 \).

Would you like more details on any of these steps, or do you have any questions?

Here are 8 related questions that expand on this information:
1. Can fixed point iteration be used to find all roots of a polynomial equation?
2. What are the conditions for the fixed point iteration method to converge?
3. How can we choose a good initial guess for the fixed point iteration method?
4. Are there other rearrangements of \( x^3 - 2x^2 - 5x + 6 = 0 \) that might work better for fixed point iteration?
5. What happens if the fixed point iteration method does not converge?
6. Can we use fixed point iteration for non-polynomial equations?
7. How do we determine the rate of convergence for the fixed point iteration method?
8. What are the advantages and disadvantages of the fixed point iteration method compared to other numerical methods for finding roots?

**Tip:** When using the fixed point iteration method, always verify the root by substituting it back into the original equation to ensure its correctness.

Learn how to find the root of the polynomial equation x³ - 2x² - 5x + 6 using the fixed point iteration method. Follow a detailed step-by-step solution including the iteration formula and verification steps. Suitable for advanced high school students and anyone interested in numerical methods for root finding.

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