(x^3-x^2-x+1)/(x^3-x^2+x-1)=0 Solve for all real values of x that satisfies 

To solve the equation:

\[
\frac{x^3 - x^2 - x + 1}{x^3 - x^2 + x - 1} = 0
\]

We need to find when the numerator is zero, because a fraction equals zero only when its numerator is zero (assuming the denominator is not zero).

### Step 1: Solve the numerator
The numerator is \(x^3 - x^2 - x + 1\). We need to find when this expression equals zero:
\[
x^3 - x^2 - x + 1 = 0
\]

We can attempt factoring by grouping. Group the terms as follows:
\[
(x^3 - x^2) - (x - 1)
\]
Factor each group:
\[
x^2(x - 1) - 1(x - 1)
\]
Now factor out \((x - 1)\):
\[
(x - 1)(x^2 - 1) = 0
\]

Notice that \(x^2 - 1\) is a difference of squares, so we can factor it further:
\[
(x - 1)(x - 1)(x + 1) = 0
\]

Thus, the factored form of the numerator is:
\[
(x - 1)^2(x + 1) = 0
\]

### Step 2: Solve for \(x\)
From the factored form, set each factor equal to zero:
\[
(x - 1)^2 = 0 \quad \text{or} \quad (x + 1) = 0
\]

This gives the solutions:
\[
x = 1 \quad \text{(double root)} \quad \text{or} \quad x = -1
\]

### Step 3: Check the denominator
The denominator is \(x^3 - x^2 + x - 1\). For the solutions \(x = 1\) and \(x = -1\) to be valid, the denominator must not be zero.

- When \(x = 1\):
  \[
  1^3 - 1^2 + 1 - 1 = 1 - 1 + 1 - 1 = 0
  \]
  So, \(x = 1\) makes the denominator zero, which means \(x = 1\) is not a valid solution.

- When \(x = -1\):
  \[
  (-1)^3 - (-1)^2 + (-1) - 1 = -1 - 1 - 1 - 1 = -4
  \]
  The denominator is \(-4 \neq 0\), so \(x = -1\) is a valid solution.

### Final Solution
The only real solution to the equation is:
\[
x = -1
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What happens if both the numerator and the denominator are zero at a specific \(x\)?
2. How do you factor cubic equations in general?
3. Can you solve this equation using the Rational Root Theorem?
4. How do double roots affect the behavior of a graph?
5. What are the implications of the difference of squares in factoring?

**Tip:** Always check for common roots between the numerator and the denominator, as they may cancel out or make a solution invalid.

Solve for all real values of x that satisfy the equation (x^3 - x^2 - x + 1)/(x^3 - x^2 + x - 1) = 0

Explore the step-by-step solution for solving the cubic equation (x^3 - x^2 - x + 1)/(x^3 - x^2 + x - 1) = 0. Learn key algebraic concepts like factoring by grouping and the zero product property, with a focus on real solutions for x. Suitable for high school students in Grades 10-12.

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