Bijection from set of equivalence classes to $\mathbb R$

In $\mathbb R[X]$, define an equivalence relation ~ by $P_1$~$P_2$ if $P_1-P_2$ is divisible by X.

I have shown that $X$ is an equivalence relation.

Let $\mathscr Q$ denote the set of equivalence classes of ~ in $\mathbb R[X]$.

I now have to find an explicit bijection $\mathscr Q \to \mathbb R$. Any help would be appreciated. Thanks.

Solution 1:

Hint:

Define a map

$$\phi:\Bbb R[x]\to\Bbb R\;,\;\;\phi(p(x)):=p(0)$$

The nicest thing of the above is that $\;\phi\;$ is a ring homomorphism...

Solution 2:

We have $P_1\sim P_2\iff x|(P_1-P_2)\iff \exists Q$ such that $P_1(x)=xQ(x)+P_2$ and by the Euclidean division we have $$P_1(x)=xQ(x)+P_1(0)$$ hence $$P_1\sim P_1(0)$$ Can you see now the bijection?