Find all the zeros of the function in the x-interval [−2,2]. Separate multiple answers with a comma and round to two decimal places, if necessary. If there are no zeros in the given interval, write NA for your answer.
 
f(x)=x^3−x^2+x+1


To find the zeros of \( f(x) = x^3 - x^2 + x + 1 \) in the interval \( [-2, 2] \), we need to solve \( f(x) = 0 \).

### Steps:
1. **Check for roots in the interval** by identifying approximate locations of zeros using numerical techniques.
2. Use a more precise method (e.g., Newton's method or bisection) to compute the zeros to two decimal places.

The function is given as:
\[
f(x) = x^3 - x^2 + x + 1.
\]

We will evaluate this function and determine the zeros.

### Calculations:
Let’s numerically approximate the zeros. numpy as np
from scipy.optimize import root_scalar

# Define the function
def f(x):
    return x**3 - x**2 + x + 1

# Interval for finding zeros
x_interval = [-2, 2]

# Find zeros using root_scalar in the interval
results = []
for i in range(x_interval[0], x_interval[1]):
    try:
        sol = root_scalar(f, bracket=[i, i + 1], method='bisect')
        if sol.converged:
            zero = round(sol.root, 2)
            if zero not in results:  # Avoid duplicates
                results.append(zero)
    except ValueError:
        continue

resultsThe zero of the function \( f(x) = x^3 - x^2 + x + 1 \) in the interval \([-2, 2]\) is approximately:

\[
x = -0.54
\]

Would you like me to verify or explain this further?

---

### Related Questions:
1. How do you derive the roots of a cubic polynomial analytically?
2. Can there be a graphical method to confirm the roots of \( f(x) \)?
3. What happens if we expand the interval \([-2, 2]\) to include a larger range?
4. How does the derivative \( f'(x) \) relate to the behavior of the roots?
5. Could other numerical methods (e.g., Newton-Raphson) yield faster convergence?

### Tip:
For precise root calculations, always verify convergence and uniqueness of solutions in the chosen interval!

Find all the zeros of the function in the x-interval [−2,2]. Separate multiple answers with a comma and round to two decimal places, if necessary. If there are no zeros in the given interval, write NA for your answer. f(x)=x^3−x^2+x+1

Learn how to find the zeros of the cubic function f(x) = x^3 - x^2 + x + 1 in the interval [-2, 2]. This solution uses numerical methods like the bisection method to approximate the root at x ≈ -0.54. Suitable for high school students in Grades 11-12.

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