## Simplifying the Expression (x^3 - 1) / (x^2 - 1)

The expression (x^3 - 1) / (x^2 - 1) can be simplified using factorization and cancellation. Let's break down the steps:

### Factorization

**1. Difference of Cubes:** The numerator (x^3 - 1) is a difference of cubes, which factors into:
**(a^3 - b^3) = (a - b)(a^2 + ab + b^2)**

In this case, a = x and b = 1, so:
**(x^3 - 1) = (x - 1)(x^2 + x + 1)**

**2. Difference of Squares:** The denominator (x^2 - 1) is a difference of squares, which factors into:
**(a^2 - b^2) = (a + b)(a - b)**

In this case, a = x and b = 1, so:
**(x^2 - 1) = (x + 1)(x - 1)**

### Cancellation

Now we have:
**(x - 1)(x^2 + x + 1) / (x + 1)(x - 1)**

Notice that (x - 1) appears in both the numerator and denominator. We can cancel these terms, leaving us with:

**(x^2 + x + 1) / (x + 1)**

### Restrictions

It's important to note that the original expression is undefined when the denominator is zero. This occurs when x = 1 and x = -1. Therefore, the simplified expression **(x^2 + x + 1) / (x + 1)** is valid for all values of x except for x = -1 and x = 1.

### Conclusion

By factoring the numerator and denominator and cancelling common factors, we simplified the expression (x^3 - 1) / (x^2 - 1) to **(x^2 + x + 1) / (x + 1)**, valid for all x except x = -1 and x = 1.