$29 \cdot 41$ If $3 x^{3}-9 x^{2}+k x-12$ is divisible by $x-3$, then it is also divisible by
(A) $3 x^{2}-x+4$.
(B) $3 x^{2}-4$
(C) $3 x^{2}+4$.
(D) $3 x-4$.
(E) $3 x+4$.
(12th American High School Mathematics Examination, 1961)
[Solution] Given $f(x)=3 x^{3}-9 x^{2}+k x-12$ can be divided by $x-3$, then $f(3)=0$, i.e., $3 \cdot 3^{3}-9 \cdot 3^{2}+3 k-12=0$, we get $k=4$. At this point, $\left(3 x^{3}-9 x^{2}+4 x-12\right) \div(x-3)=3 x^{2}+4$. Therefore, the answer is $(C)$.