For $r(x)=x^{2 t}$, show that $\frac{1}{x-a}=\sum_{j=1}^{2 t} a^{-j} x^{j-1}$. Use this to verify again that the Goppa code determined by $x^{2 t}$ and all nonzero elements $\alpha^0, \alpha^{-1}, \ldots, \alpha^{-(n-1)}$ of the extension field is precisely the BCH code.