Consider $f(x)=x^n-x^s-1$ and $g(x)=x^i-x^j-1$ , I want to find $Resultant(f,g)$. It is well known that it is determinant of a Sylvester matrix but, I am finding it to obscure to evaluate in that way.

Is there some known result for such special cases ?

It is known that the Discriminant of a trinomial has a closed form expression. (LINK)