To find the values of x that satisfy the equation x^3 - x^2 - x + 1 = 0, we can use various methods such as factoring, synthetic division, or numerical methods. However, in this case, the equation does not appear to factor easily. Let's use numerical methods, such as the Newton-Raphson method, to approximate the solutions.
Using an initial guess of x = 1, we can iteratively refine our approximation until we find a value that satisfies the equation.
Here's a step-by-step approach using the Newton-Raphson method:
1. Take an initial guess for x, such as x = 1.
2. Calculate the function value f(x) = x^3 - x^2 - x + 1 and its derivative f'(x) = 3x^2 - 2x - 1 at that point.
3. Apply the Newton-Raphson formula: x_new = x - f(x)/f'(x).
4. Substitute the current x value into the formula and calculate the updated x_new.
5. Repeat steps 2-4 until the value of f(x) approaches zero or the desired level of accuracy.
By repeatedly applying these steps, we can find the approximate values of x that satisfy the equation.