The term independent of `x` in the product `(4 + x + 7x^2)(x-3/x)^11` is (a) `7*"^(11)C_(6)` (b) `(3)^6*"^(11)C_(6)` (c) `3^(5)*"^(11)C_(5)` (d) `-12*2^(11)`

Options

A. `7*"^(11)C_(6)`
B. `36*"^(11)C_(6)`
C. `3^(5)*"^(11)C_(5)`
D. `-12*2^(11)`

Explanation

To find the term independent of \( x \) in the expression \( (4 + x + 7x^2)(x - \frac{3}{x})^{11} \), we will break down the problem step by step.

### Step 1: Expand the Binomial Expression
We start with the binomial expression \( (x - \frac{3}{x})^{11} \). According to the Binomial Theorem, the general term (the \( r+1 \)th term) in the expansion of \( (a + b)^n \) is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

In our case, \( a = x \), \( b = -\frac{3}{x} \), and \( n = 11 \). Thus, the general term becomes:

\[ T_{r+1} = \binom{11}{r} x^{11-r} \left(-\frac{3}{x}\right)^r = \binom{11}{r} (-3)^r x^{11 - 2r} \]

### Step 2: Identify the Terms from the First Factor
Now we need to multiply this general term by each term in the first factor \( (4 + x + 7x^2) \):

1. **From \( 4 \)**:
\[ 4 \cdot T_{r+1} = 4 \cdot \binom{11}{r} (-3)^r x^{11 - 2r} \]

2. **From \( x \)**:
\[ x \cdot T_{r+1} = \binom{11}{r} (-3)^r x^{11 - 2r + 1} \]

3. **From \( 7x^2 \)**:
\[ 7x^2 \cdot T_{r+1} = 7 \cdot \binom{11}{r} (-3)^r x^{11 - 2r + 2} \]

### Step 3: Find the Condition for Independence from \( x \)
We need to find the value of \( r \) such that the power of \( x \) is zero in each case.

1. **From \( 4 \cdot T_{r+1} \)**:
\[ 11 - 2r = 0 \implies 2r = 11 \implies r = \frac{11}{2} \quad (\text{not an integer}) \]

2. **From \( x \cdot T_{r+1} \)**:
\[ 11 - 2r + 1 = 0 \implies 12 - 2r = 0 \implies 2r = 12 \implies r = 6 \]

3. **From \( 7x^2 \cdot T_{r+1} \)**:
\[ 11 - 2r + 2 = 0 \implies 13 - 2r = 0 \implies 2r = 13 \implies r = \frac{13}{2} \quad (\text{not an integer}) \]

### Step 4: Calculate the Independent Term
The only valid \( r \) is \( r = 6 \). Now we substitute \( r = 6 \) into the general term:

\[ T_{r+1} = \binom{11}{6} (-3)^6 \]

Thus, the term independent of \( x \) from \( 4 \cdot T_{r+1} \) is:

\[ 4 \cdot \binom{11}{6} (-3)^6 \]

### Step 5: Simplify the Expression
Calculating \( (-3)^6 \):

\[ (-3)^6 = 729 \]

Thus, the term becomes:

\[ 4 \cdot \binom{11}{6} \cdot 729 \]

### Final Answer
The term independent of \( x \) is:

\[ 4 \cdot \binom{11}{6} \cdot 729 \]