Question

The value of \( \left( x - \frac{2}{x} \right) \left( x^2 + 2 + \frac{4}{x^2} \right) \) is equal to:

• A. \( x^3 + 2x + \frac{4}{x} - 8 \)
• B. \( x^3 - \frac{8}{x^3} \)
• C. \( x^3 + \frac{8}{x^3} \)
• D. \( x^3 - \frac{8}{x^2} \)

Hint:

When multiplying binomials, distribute each term across the other binomial, then simplify the resulting terms.

Answer 1

The Correct Option is A

We are asked to find the value of: \[ \left( x - \frac{2}{x} \right) \left( x^2 + 2 + \frac{4}{x^2} \right) \] We can distribute \( \left( x - \frac{2}{x} \right) \) across \( \left( x^2 + 2 + \frac{4}{x^2} \right) \): \[ = x(x^2 + 2 + \frac{4}{x^2}) - \frac{2}{x}(x^2 + 2 + \frac{4}{x^2}) \] Simplifying each term: \[ = x^3 + 2x + \frac{4}{x} - \frac{2x^2}{x} - \frac{4}{x} \] \[ = x^3 + 2x + \frac{4}{x} - 8 \] Thus, the correct answer is \( x^3 + 2x + \frac{4}{x} - 8 \).