## Factoring and Simplifying the Expression (w^3 - 2w^2 - 2w + 1) / (w - 1)

This article explores the process of factoring and simplifying the rational expression (w^3 - 2w^2 - 2w + 1) / (w - 1).

### Understanding the Expression

The given expression is a rational expression, which is a fraction where both the numerator and denominator are polynomials. Our goal is to simplify this expression, if possible.

### Factoring the Numerator

We begin by factoring the numerator, w^3 - 2w^2 - 2w + 1. We can use the following steps:

**Grouping:**Group the terms in the numerator: (w^3 - 2w^2) + (-2w + 1).**Factoring out common factors:**Factor out w^2 from the first group and -1 from the second group: w^2(w - 2) - 1(w - 2).**Factoring out the common binomial:**Notice that (w - 2) is common to both terms. Factor it out: (w - 2)(w^2 - 1).**Factoring the difference of squares:**The term (w^2 - 1) is a difference of squares, which can be factored as (w + 1)(w - 1).

Therefore, the factored numerator is **(w - 2)(w + 1)(w - 1)**.

### Simplifying the Expression

Now we can rewrite the original expression:

(w^3 - 2w^2 - 2w + 1) / (w - 1) = [(w - 2)(w + 1)(w - 1)] / (w - 1)

Since (w - 1) appears in both the numerator and denominator, we can cancel them out:

[(w - 2)(w + 1)(w - 1)] / (w - 1) = **(w - 2)(w + 1)**

### Conclusion

The simplified form of the expression (w^3 - 2w^2 - 2w + 1) / (w - 1) is **(w - 2)(w + 1)**. This simplification is valid for all values of w except for w = 1, where the original expression is undefined.