AHSME 1961 Problem 22

If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: $\textbf{(A)}\ 3x^2-x+4 \textbf{(B)}\ 3x^2-4 \textbf{(C)}\ 3x^2+4 \textbf{(D)}\ 3x-4 \textbf{(E)}\ 3x+4$