If `3x ^(4) -6x ^(3) +kx ^(2)-8x-12`is divisible by `x-3,` then it is also divisible by :

Options

A. `3x ^(2) -4`
B. `3x ^(2) + 4`
C. `3x ^(2) +x`
D. `3x ^(2)-x `

Answer

To solve the problem, we need to determine the value of \( k \) such that the polynomial \( 3x^4 - 6x^3 + kx^2 - 8x - 12 \) is divisible by \( x - 3 \).

### Step-by-Step Solution:

1. **Set Up the Equation**:
Since the polynomial is divisible by \( x - 3 \), we can substitute \( x = 3 \) into the polynomial and set it equal to zero.
\[ 3(3^4) - 6(3^3) + k(3^2) - 8(3) - 12 = 0 \]

2. **Calculate Each Term**:
- Calculate \( 3^4 = 81 \), so \( 3(3^4) = 3 \times 81 = 243 \).
- Calculate \( 3^3 = 27 \), so \( -6(3^3) = -6 \times 27 = -162 \).
- Calculate \( 3^2 = 9 \), so \( k(3^2) = 9k \).
- Calculate \( -8(3) = -24 \).
- The constant term is \( -12 \).

Now substituting these values into the equation:
\[ 243 - 162 + 9k - 24 - 12 = 0 \]

3. **Combine Like Terms**:
Combine the constant terms:
\[ 243 - 162 - 24 - 12 = 45 \]
So the equation simplifies to:
\[ 45 + 9k = 0 \]

4. **Solve for \( k \)**:
Rearranging gives:
\[ 9k = -45 \]
Dividing both sides by 9:
\[ k = -5 \]

5. **Rewrite the Polynomial**:
Now that we have \( k \), we can rewrite the polynomial:
\[ 3x^4 - 6x^3 - 5x^2 - 8x - 12 \]

6. **Perform Polynomial Long Division**:
To find other factors, we will divide the polynomial by \( x - 3 \):
\[ \text{Divide } 3x^4 - 6x^3 - 5x^2 - 8x - 12 \text{ by } x - 3 \]

- The first term is \( 3x^3 \).
- Multiply \( 3x^3 \) by \( x - 3 \) to get \( 3x^4 - 9x^3 \).
- Subtract: \( (-6x^3 + 9x^3) = 3x^3 \).
- Bring down the next term: \( 3x^3 - 5x^2 \).
- The next term is \( 3x^2 \).
- Repeat the process until the remainder is 0.

7. **Final Factors**:
After performing the division, we find:
\[ 3x^3 + 3x^2 + 4x + 4 \]
We can factor this further.

8. **Factorization**:
The polynomial can be factored as:
\[ (x - 3)(3x^2 + 4)(x + 1) \]
Therefore, the polynomial is divisible by \( x + 1 \) and \( 3x^2 + 4 \).

### Final Answer:
The polynomial \( 3x^4 - 6x^3 - 5x^2 - 8x - 12 \) is also divisible by \( x + 1 \) and \( 3x^2 + 4 \).