%2F(w-1).jpeg "(w^3-2w^2-2w+1)/(w-1)")
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.