Simplify (1/x - x)²: Square of Difference Expression

To solve this problem, we'll use the square of a binomial formula:

• Identify $a = \frac{1}{x}$ and $b = x$.
• Apply the formula $(a - b)^2 = a^2 - 2ab + b^2$.
• Simplify each component and obtain the final expression.

Let's work through the steps:

Step 1: Identify the components:
$a = \frac{1}{x}$, $b = x$

Step 2: Apply the formula:

$(\frac{1}{x} - x)^2 = (\frac{1}{x})^2 - 2 \times \frac{1}{x} \times x + x^2$

Simplifying each term:

$(\frac{1}{x})^2 = \frac{1}{x^2}$

$-2 \times \frac{1}{x} \times x = -2$

$x^2 = x^2$

Step 3: Combine and simplify:

$\frac{1}{x^2} - 2 + x^2$

To combine these into a single fraction, find a common denominator $x^2$:

$\frac{1}{x^2} - \frac{2x^2}{x^2} + \frac{x^4}{x^2} = \frac{1 - 2x^2 + x^4}{x^2}$

Thus, the simplified expression is:

$\frac{x^4 - 2x^2 + 1}{x^2}$

Comparing to the choices provided, the correct choice is:

Choice 3: $\frac{x^4 - 2x^2 + 1}{x^2}$