Find the term independent of `x` in the expansion of `(1+x+2x^3)[(3x^2//2)-(1//3)]^9`

Options

A. `(1)/(3)`
B. `(19)/(54)`
C. `(17)/(54)`
D. `(1)/(4)`

Answer

To find the term independent of \( x \) in the expansion of \( (1 + x + 2x^3) \left( \frac{3x^2}{2} - \frac{1}{3} \right)^9 \), we will follow these steps:

### Step 1: Identify the General Term
The general term in the expansion of \( \left( \frac{3x^2}{2} - \frac{1}{3} \right)^9 \) can be expressed using the binomial theorem:
\[ T_r = \binom{9}{r} \left( \frac{3x^2}{2} \right)^{9-r} \left( -\frac{1}{3} \right)^r \]
This simplifies to:
\[ T_r = \binom{9}{r} \cdot \frac{3^{9-r}}{2^{9-r}} \cdot (-1)^r \cdot x^{2(9-r)} \]

### Step 2: Simplify the General Term
Now, we can rewrite the general term as:
\[ T_r = \binom{9}{r} \cdot \frac{3^{9-r}}{2^{9-r}} \cdot (-1)^r \cdot x^{18 - 2r} \]

### Step 3: Multiply by \( (1 + x + 2x^3) \)
Next, we need to consider the multiplication with \( (1 + x + 2x^3) \). This means we will have three cases to consider:
1. Multiplying by \( 1 \)
2. Multiplying by \( x \)
3. Multiplying by \( 2x^3 \)

### Step 4: Find Terms Independent of \( x \)
We need to find the values of \( r \) such that the total power of \( x \) becomes zero.

1. **From multiplying by \( 1 \):**
\[ 18 - 2r = 0 \implies r = 9 \]

2. **From multiplying by \( x \):**
\[ 19 - 2r = 0 \implies r = \frac{19}{2} \text{ (not an integer, discard)} \]

3. **From multiplying by \( 2x^3 \):**
\[ 21 - 2r = 0 \implies r = \frac{21}{2} \text{ (not an integer, discard)} \]

The only valid solution is \( r = 9 \).

### Step 5: Calculate the Independent Term
Substituting \( r = 9 \) into the general term:
\[ T_9 = \binom{9}{9} \cdot \frac{3^{0}}{2^{0}} \cdot (-1)^9 \cdot x^{0} = 1 \cdot 1 \cdot (-1) \cdot 1 = -1 \]

### Step 6: Conclusion
Thus, the term independent of \( x \) in the expansion is:
\[ \text{Independent term} = -\frac{1}{3^9} \]