Understanding a ring homomorphism

Question

By definition, we have that:

$\mathbb{Q}[x,y] = (\mathbb{Q}[x])[y] = (\mathbb{Q}[y])[x]$.

Now I want to come to the conclusion that: $(\mathbb{Q}[y])[x]/(x) \cong \mathbb{Q}[y]$.

I'm not sure how to go about

Answer 1

Hint: Let $R:=\Bbb Q[y]$, and determine the kernel of the homomorphism $R[x]\to R$ sending $x\mapsto0$.

Answer 2

Hint:

You can define the homomorphism as

\begin{align} \mathbf Q[y][x]&\longrightarrow\mathbf Q[y]\\ P(x,y)&\longmapsto P(0,y) \end{align}