Expand `((2)/( x) - (x)/(2))^(5), x ne 0`.

Answer

To expand the expression \(\left(\frac{2}{x} - \frac{x}{2}\right)^{5}\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify the terms in the binomial expression
The expression can be rewritten as:
\[ a = \frac{2}{x}, \quad b = -\frac{x}{2} \]
We will use the Binomial Theorem which states:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, \(n = 5\).

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem:
\[ \left(\frac{2}{x} - \frac{x}{2}\right)^{5} = \sum_{k=0}^{5} \binom{5}{k} \left(\frac{2}{x}\right)^{5-k} \left(-\frac{x}{2}\right)^{k} \]

### Step 3: Calculate each term in the expansion
Now we will calculate each term for \(k = 0\) to \(k = 5\):

- **For \(k = 0\)**:
\[ \binom{5}{0} \left(\frac{2}{x}\right)^{5} \left(-\frac{x}{2}\right)^{0} = 1 \cdot \frac{32}{x^5} \cdot 1 = \frac{32}{x^5} \]

- **For \(k = 1\)**:
\[ \binom{5}{1} \left(\frac{2}{x}\right)^{4} \left(-\frac{x}{2}\right)^{1} = 5 \cdot \frac{16}{x^4} \cdot \left(-\frac{x}{2}\right) = -\frac{80}{x^3} \]

- **For \(k = 2\)**:
\[ \binom{5}{2} \left(\frac{2}{x}\right)^{3} \left(-\frac{x}{2}\right)^{2} = 10 \cdot \frac{8}{x^3} \cdot \frac{x^2}{4} = \frac{80}{x} \]

- **For \(k = 3\)**:
\[ \binom{5}{3} \left(\frac{2}{x}\right)^{2} \left(-\frac{x}{2}\right)^{3} = 10 \cdot \frac{4}{x^2} \cdot \left(-\frac{x^3}{8}\right) = -\frac{40x}{2} = -20x \]

- **For \(k = 4\)**:
\[ \binom{5}{4} \left(\frac{2}{x}\right)^{1} \left(-\frac{x}{2}\right)^{4} = 5 \cdot \frac{2}{x} \cdot \frac{x^4}{16} = \frac{10x^3}{16} = \frac{5x^3}{8} \]

- **For \(k = 5\)**:
\[ \binom{5}{5} \left(\frac{2}{x}\right)^{0} \left(-\frac{x}{2}\right)^{5} = 1 \cdot 1 \cdot \left(-\frac{x^5}{32}\right) = -\frac{x^5}{32} \]

### Step 4: Combine all the terms
Now, we combine all the terms:
\[ \frac{32}{x^5} - \frac{80}{x^3} + \frac{80}{x} - 20x + \frac{5x^3}{8} - \frac{x^5}{32} \]

### Final Answer
Thus, the expansion of \(\left(\frac{2}{x} - \frac{x}{2}\right)^{5}\) is:
\[ \frac{32}{x^5} - \frac{80}{x^3} + \frac{80}{x} - 20x + \frac{5x^3}{8} - \frac{x^5}{32} \]