## Simplifying the Rational Expression: (x^4 - x^2) / (x^3 - 2x^2 + x)

This article explores the simplification of the rational expression (x^4 - x^2) / (x^3 - 2x^2 + x). We'll break down the steps to arrive at a simplified form.

### Step 1: Factor the numerator and denominator

Both the numerator and denominator can be factored:

**Numerator:**x^4 - x^2 = x^2(x^2 - 1) = x^2(x+1)(x-1)**Denominator:**x^3 - 2x^2 + x = x(x^2 - 2x + 1) = x(x-1)^2

### Step 2: Simplify by canceling common factors

Now we have:

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

Notice that x and (x-1) appear in both the numerator and denominator. We can cancel these out:

x(x+1) / (x-1)

### Step 3: The simplified expression

The simplified form of the rational expression is **x(x+1) / (x-1)**.

**Important Notes:**

- We cannot simplify further as there are no common factors remaining.
- This simplified expression is equivalent to the original expression for all values of x except x = 0 and x = 1. These are the values that make the denominator of the original expression zero, leading to undefined expressions.

This process demonstrates how factoring can be used to simplify rational expressions. By identifying common factors and canceling them, we can often express complex expressions in a more concise and manageable form.