Determine an equivalence class of an element

Question

Be $ R $ a relation in $ \mathbb{Q\left[x\right]} $ such that $ \mathcal{F}R\,\mathcal{G} \iff x^3 - x \vert\, x^{3}\mathcal{F} - x\mathcal{G} $

a) Show it is an equivalence relation

b) Determine the equivalence class for $\mathcal{F} = 1$

Now, I was able to demonstrate it is reflexive, symmetric and transitive but I can't see how to approach point b).

I understand that this equivalence class is the set $ \{\mathcal{G} \in \mathbb{Q\left[x\right]} \, / x^3 - x \vert x^3 - x\mathcal{G}\} $

Answer 1

As you said, they are element s.t. $$x^3-x\mathcal G=P(x)(x^3-x)\iff\mathcal G=P(x)(1-x^2)+x^2=Q(x)(1-x^2)+1,$$ thus the class of $\mathcal F=1$ are elements that has remainder $1$ in division by $1-x^2$.

Answer 2

Hint:

$x^3-x\mid P(x)\iff \bigl(P(0)=0\enspace\text{and}\enspace P(1)=0\enspace\text{and}\enspace P(-1)=0\bigl)$.

Translate these conditions into the division of $P(x)=1-\mathcal G(x)$.