On the Diophantine equation $f(x)f(y)=f(z{)}^n$ involving Laurent polynomials

By the theory of elliptic curves, we investigate the nontrivial rational parametric solutions of the Diophantine equation $f(x)f(y)=f(z{)}^n$, where $n=1,2$ and $f(X)$ are some simple Laurent polynomials.