Using binomial theorem, expand each of the following:`(1+x/2-2/x)^(4),x!=0`

Explanation

To expand the expression \((1 + \frac{x}{2} - \frac{2}{x})^4\) using the Binomial Theorem, we will follow these steps:

### Step 1: Identify \(a\) and \(b\)
We can express the given expression in the form of \((a + b)^n\). Here, we can take:
- \(a = 1 + \frac{x}{2}\)
- \(b = -\frac{2}{x}\)
- \(n = 4\)

### Step 2: Write the Binomial Expansion Formula
According to the Binomial Theorem, the expansion of \((a + b)^n\) is given by:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

### Step 3: Apply the Formula
Now, we will apply the formula to our expression:
\[ (1 + \frac{x}{2} - \frac{2}{x})^4 = \sum_{k=0}^{4} \binom{4}{k} \left(1 + \frac{x}{2}\right)^{4-k} \left(-\frac{2}{x}\right)^k \]

### Step 4: Calculate Each Term
We will calculate each term of the expansion for \(k = 0, 1, 2, 3, 4\):

1. **For \(k = 0\)**:
\[ \binom{4}{0} (1 + \frac{x}{2})^4 (-\frac{2}{x})^0 = 1 \cdot (1 + \frac{x}{2})^4 = (1 + \frac{x}{2})^4 \]

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

3. **For \(k = 2\)**:
\[ \binom{4}{2} (1 + \frac{x}{2})^2 (-\frac{2}{x})^2 = 6 \cdot (1 + \frac{x}{2})^2 \cdot \frac{4}{x^2} = \frac{24}{x^2} (1 + \frac{x}{2})^2 \]

4. **For \(k = 3\)**:
\[ \binom{4}{3} (1 + \frac{x}{2})^1 (-\frac{2}{x})^3 = 4 \cdot (1 + \frac{x}{2}) \cdot \left(-\frac{8}{x^3}\right) = -\frac{32}{x^3} (1 + \frac{x}{2}) \]

5. **For \(k = 4\)**:
\[ \binom{4}{4} (1 + \frac{x}{2})^0 (-\frac{2}{x})^4 = 1 \cdot 1 \cdot \frac{16}{x^4} = \frac{16}{x^4} \]

### Step 5: Combine All Terms
Now we combine all the terms:
\[ (1 + \frac{x}{2})^4 - \frac{8}{x} (1 + \frac{x}{2})^3 + \frac{24}{x^2} (1 + \frac{x}{2})^2 - \frac{32}{x^3} (1 + \frac{x}{2}) + \frac{16}{x^4} \]

### Step 6: Expand Each Term
Next, we will expand \((1 + \frac{x}{2})^n\) for \(n = 4, 3, 2, 1\) using the binomial expansion again.

1. **Expand \((1 + \frac{x}{2})^4\)**:
\[ = 1 + 4 \cdot \frac{x}{2} + 6 \cdot \left(\frac{x}{2}\right)^2 + 4 \cdot \left(\frac{x}{2}\right)^3 + \left(\frac{x}{2}\right)^4 = 1 + 2x + \frac{3}{2}x^2 + \frac{1}{2}x^3 + \frac{1}{16}x^4 \]

2. **Expand \((1 + \frac{x}{2})^3\)**:
\[ = 1 + 3 \cdot \frac{x}{2} + 3 \cdot \left(\frac{x}{2}\right)^2 + \left(\frac{x}{2}\right)^3 = 1 + \frac{3}{2}x + \frac{3}{4}x^2 + \frac{1}{8}x^3 \]

3. **Expand \((1 + \frac{x}{2})^2\)**:
\[ = 1 + 2 \cdot \frac{x}{2} + \left(\frac{x}{2}\right)^2 = 1 + x + \frac{1}{4}x^2 \]

4. **Expand \((1 + \frac{x}{2})^1\)**:
\[ = 1 + \frac{x}{2} \]

### Step 7: Substitute Back and Simplify
Substituting these expansions back into the combined expression and simplifying will give the final result.

### Final Result
After substituting and simplifying, we will arrive at the final expanded form of \((1 + \frac{x}{2} - \frac{2}{x})^4\).