1/x+xcuadrado - 1/xcuadrado-x - x+3/xcuadrado-1

Let's break down and simplify the expression step by step:

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

### Step 1: Combine like terms.
We will now group the terms with the same variable powers.

\[
\left( \frac{1}{x} - \frac{1}{x^2} + \frac{3}{x^2} \right) + \left( x^2 - x - x - 1 \right)
\]

### Step 2: Simplify fractions and powers.
For the fractions:

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

For the rest of the terms:

\[
x^2 - 2x - 1
\]

### Step 3: Combine all simplified terms.
So, the final simplified expression is:

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

This is the simplified form of the given expression.

Would you like a more detailed breakdown or explanation? Here are some additional questions related to this expression:

1. What happens to the expression when \( x = 1 \)?
2. Can this expression be factored further?
3. How does the expression behave for large values of \( x \)?
4. What is the domain of this expression?
5. How does the expression behave as \( x \to 0 \)?

**Tip:** Always check for common denominators when simplifying expressions involving fractions.

This page provides a detailed, step-by-step solution to simplifying the expression 1/x + x^2 - 1/x^2 - x + 3/x^2 - 1. It covers key algebraic concepts such as rational expressions, combining like terms, and polynomial simplification. Suitable for students in Grades 9-11.

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