Expand `(3x^2- x/2)^5`

Answer

To expand the expression \((3x^2 - \frac{x}{2})^5\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify the terms
In the expression \((a - b)^n\), we have:
- \(a = 3x^2\)
- \(b = \frac{x}{2}\)
- \(n = 5\)

### Step 2: Write the Binomial Expansion
According to the Binomial Theorem, the expansion of \((a - b)^n\) is given by:
\[ (a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k \]
For our case, it becomes:
\[ (3x^2 - \frac{x}{2})^5 = \sum_{k=0}^{5} \binom{5}{k} (3x^2)^{5-k} \left(-\frac{x}{2}\right)^k \]

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

1. **For \(k = 0\)**:
\[ \binom{5}{0} (3x^2)^5 \left(-\frac{x}{2}\right)^0 = 1 \cdot (3x^2)^5 = 243x^{10} \]

2. **For \(k = 1\)**:
\[ \binom{5}{1} (3x^2)^4 \left(-\frac{x}{2}\right)^1 = 5 \cdot (3x^2)^4 \cdot \left(-\frac{x}{2}\right) = -\frac{15 \cdot 81 x^9}{2} = -\frac{1215x^9}{2} \]

3. **For \(k = 2\)**:
\[ \binom{5}{2} (3x^2)^3 \left(-\frac{x}{2}\right)^2 = 10 \cdot (3x^2)^3 \cdot \left(\frac{x^2}{4}\right) = 10 \cdot 27x^6 \cdot \frac{x^2}{4} = \frac{270x^8}{4} = 67.5x^8 \]

4. **For \(k = 3\)**:
\[ \binom{5}{3} (3x^2)^2 \left(-\frac{x}{2}\right)^3 = 10 \cdot (3x^2)^2 \cdot \left(-\frac{x^3}{8}\right) = 10 \cdot 9x^4 \cdot \left(-\frac{x^3}{8}\right) = -\frac{90x^7}{8} = -11.25x^7 \]

5. **For \(k = 4\)**:
\[ \binom{5}{4} (3x^2)^1 \left(-\frac{x}{2}\right)^4 = 5 \cdot (3x^2) \cdot \left(\frac{x^4}{16}\right) = 5 \cdot 3x^2 \cdot \frac{x^4}{16} = \frac{15x^6}{16} \]

6. **For \(k = 5\)**:
\[ \binom{5}{5} (3x^2)^0 \left(-\frac{x}{2}\right)^5 = 1 \cdot 1 \cdot \left(-\frac{x^5}{32}\right) = -\frac{x^5}{32} \]

### Step 4: Combine all the terms
Now we combine all the terms we calculated:
\[ (3x^2 - \frac{x}{2})^5 = 243x^{10} - \frac{1215x^9}{2} + 67.5x^8 - 11.25x^7 + \frac{15x^6}{16} - \frac{x^5}{32} \]

### Final Answer
Thus, the expansion of \((3x^2 - \frac{x}{2})^5\) is:
\[ 243x^{10} - \frac{1215}{2}x^9 + 67.5x^8 - 11.25x^7 + \frac{15}{16}x^6 - \frac{1}{32}x^5 \]