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 x3 - x2 + 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?

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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!