Prove that $\mathbb{Q}[X]/I\cong Q\times Q$

Question

Let $f(X)=(X^2-2)(X^4-X)$ and $g(X)=(X^2-1)X\in \mathbb{Q}[X]$. Let $I=(f,g)$ the ideal generated by $f$ and $g$. Prove that $\mathbb{Q}[X]/I\cong Q\times Q$

Using the reasoning of this answer I have computed that $I=X(X-1)$.

The next step is to define an epimorphism $\phi: \mathbb{Q}[X]\longrightarrow Q\times Q$, see that $\ker \phi = I$ and apply the First Isomorphism Theorem. But I am not sure about what is $Q$, I think that it is the field of fractions of $\mathbb{Q}[X]$ and I am not clear about how to define $\phi$.

Answer 1

Hint the mapping $P \to P(a)$ has kernel $\langle x-a \rangle$.

So try to set $\phi(P)= (P(\alpha), P(\beta))$ in such a way that the kernel becomes exactly $$\langle X(X-1) \rangle= \langle X\rangle \cap \langle X-1 \rangle$$