If `3x^(2) +ax +12` is perfectly divisible by x -3 then the value of a is
A)`-13`
B)`-12`
C)`-3 `
D)`-9`

Options

A. `-13`
B. `-12`
C. `-3 `
D. `-9`

Answer

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

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- The polynomial is \( P(x) = 3x^2 + ax + 12 \).
- The divisor is \( x - 3 \), which means \( c = 3 \).

2. **Apply the Remainder Theorem**:
- According to the Remainder Theorem, we need to evaluate \( P(3) \) and set it equal to 0:
\[ P(3) = 3(3)^2 + a(3) + 12 \]

3. **Calculate \( P(3) \)**:
- Substitute \( x = 3 \):
\[ P(3) = 3(9) + 3a + 12 \]
- Simplifying this gives:
\[ P(3) = 27 + 3a + 12 = 39 + 3a \]

4. **Set the polynomial equal to zero**:
- Since \( P(3) \) must equal 0 for the polynomial to be divisible by \( x - 3 \):
\[ 39 + 3a = 0 \]

5. **Solve for \( a \)**:
- Rearranging the equation gives:
\[ 3a = -39 \]
- Dividing both sides by 3:
\[ a = -13 \]

### Conclusion:
The value of \( a \) is \( -13 \).