if `3x^(3)-9x^(2)+Kx-12` is divisible by `x-3`, then K=

Options

A. `-40`
B. `-3`
C. `3`
D. 4

Explanation

To find the value of \( K \) such that the polynomial \( 3x^3 - 9x^2 + Kx - 12 \) is divisible by \( x - 3 \), we can use the Remainder Theorem. According to the theorem, if a polynomial \( f(x) \) is divisible by \( x - a \), then \( f(a) = 0 \).

### Step-by-step Solution:

1. **Identify the Polynomial and the Root**:
Given polynomial:
\[ f(x) = 3x^3 - 9x^2 + Kx - 12 \]
We need to check for the root \( x = 3 \).

2. **Substitute \( x = 3 \) into the Polynomial**:
Substitute \( x = 3 \) into the polynomial:
\[ f(3) = 3(3)^3 - 9(3)^2 + K(3) - 12 \]

3. **Calculate Each Term**:
- Calculate \( 3(3)^3 \):
\[ 3(27) = 81 \]
- Calculate \( -9(3)^2 \):
\[ -9(9) = -81 \]
- Calculate \( K(3) \):
\[ 3K \]
- The constant term is \( -12 \).

4. **Combine the Terms**:
Now, combine all the terms:
\[ f(3) = 81 - 81 + 3K - 12 \]
This simplifies to:
\[ f(3) = 3K - 12 \]

5. **Set the Polynomial Equal to Zero**:
Since \( f(3) \) must equal zero for \( x - 3 \) to be a factor:
\[ 3K - 12 = 0 \]

6. **Solve for \( K \)**:
Rearranging the equation gives:
\[ 3K = 12 \]
Dividing both sides by 3:
\[ K = \frac{12}{3} = 4 \]

### Final Answer:
Thus, the value of \( K \) is:
\[ \boxed{4} \]