If $p(x)$ is one such polynomial.
$p_1(x)$ be such that
$p(x)-1=(x-3)p_1(x)$
$\Rightarrow p(1)-1=-2p_1(1)$
$\Rightarrow 1=-2p_1(1)$
$\Rightarrow p_1(1)=-\large\frac{1}{2}$
This is not possible as $p_1(1)$ is an integer.
Hence no such polynomial is possible.
Hence (A) is the correct answer.