Question

Expand the expression\( (\frac{2}{x} - \frac{x}{2})^5\).

Answer 1

By using Binomial Theorem, the expression (\((\frac{2}{x} - \frac{x}{2})^5\)can be expanded as:

\((\frac{2}{x} - \frac{x}{2})^5\) = \(^5 C_0(\frac{2}{x})^5 - ^5 C_1(\frac{2}{x})^4(\frac{x}{2})+ ^5 C_2(\frac{2}{x})^3(\frac{x}{2})^2 - ^5 C_3(\frac{2}{x})^2(\frac{x}{2})^3+ ^5 C_4(\frac{2}{x})(\frac{x}{2})^4 + ^5C_5(\frac{x}{2})^5\)

= \(\frac{32}{x^5} - 5(\frac{16}{x^4})(\frac{x}{2}) + 10(\frac{8}{x^3})(\frac{x^2}{4}) -10 (\frac{4}{x^2})(\frac{x^3}{8}) + 5(\frac{2}{x})(\frac{x^4}{16}) - \frac{x^5}{32}\)

= \(\frac{32}{x^5} - \frac{40}{x^3} + \frac{20}{x} - 5x + \frac{5}{8 x^3} - \frac{x^5}{32}\).