 Solution

In the cyclic quotient $R/(d)$, the kernel is formed by residue classes of polynomials $h$ such that $d\mid Xh$. If $X\nmid d$, coprimality forces $d\mid h$, so the kernel is zero. If $d=Xe$, divisibility is equivalent to $e\mid h$, and the residue classes of multiples of $e$ modulo $Xe$ form a one-dimensional $K$-space. HenceThis too is immediately readable in the cyclic case, even if zero is a repeated root. The zero-dimensional module has kernel dimension zero separately.