Write out the expansions of the following:
(f) `((2)/(x) - (x)/(2) )^(5) , x ne 0`

Answer

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

### Step 1: Identify the components for the Binomial Theorem
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, we can rewrite the expression as:
\[ \left(\frac{2}{x} + \left(-\frac{x}{2}\right)\right)^5 \]
Here, \(a = \frac{2}{x}\), \(b = -\frac{x}{2}\), and \(n = 5\).

### Step 2: Write out the expansion using the Binomial Theorem
Using the Binomial Theorem, we expand:
\[ \left(\frac{2}{x} - \frac{x}{2}\right)^5 = \sum_{k=0}^{5} \binom{5}{k} \left(\frac{2}{x}\right)^{5-k} \left(-\frac{x}{2}\right)^k \]

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

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

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

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

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

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

6. **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 terms
Now, we combine all the terms:
\[ \frac{32}{x^5} - \frac{80}{x^3} + \frac{40}{x} - 5x + \frac{5x^3}{8} - \frac{x^5}{32} \]

### Final Expansion
Thus, the final expansion of \(\left(\frac{2}{x} - \frac{x}{2}\right)^5\) is:
\[ \frac{32}{x^5} - \frac{80}{x^3} + \frac{40}{x} - 5x + \frac{5x^3}{8} - \frac{x^5}{32} \]