In the expantion of `(1+kx)^(4)` the cofficient of ` x^(3)` is 32, then the value of k is equal to

Options

A. 2
B. 4
C. 8
D. 1

Answer

To solve the problem, we need to find the value of \( k \) such that the coefficient of \( x^3 \) in the expansion of \( (1 + kx)^4 \) is equal to 32.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the expansion of \( (1 + kx)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} (1)^{n-r} (kx)^r \]
For our case, \( n = 4 \), so:
\[ T_{r+1} = \binom{4}{r} (kx)^r \]

2. **Find the Coefficient of \( x^3 \)**:
We need the term where \( r = 3 \) (since we want the coefficient of \( x^3 \)):
\[ T_{4} = \binom{4}{3} (kx)^3 \]
This simplifies to:
\[ T_{4} = \binom{4}{3} k^3 x^3 \]

3. **Calculate \( \binom{4}{3} \)**:
\[ \binom{4}{3} = 4 \]
Therefore:
\[ T_{4} = 4 k^3 x^3 \]

4. **Set the Coefficient Equal to 32**:
We know from the problem statement that the coefficient of \( x^3 \) is 32:
\[ 4 k^3 = 32 \]

5. **Solve for \( k^3 \)**:
Divide both sides by 4:
\[ k^3 = \frac{32}{4} = 8 \]

6. **Find \( k \)**:
Taking the cube root of both sides:
\[ k = \sqrt[3]{8} = 2 \]

### Final Answer:
The value of \( k \) is \( 2 \).