How can I show that $a_0x^n+...+a_n$ and $a_nx^n+...a_0$ have the same discriminant?
You can use two different definition of the discriminant of the polynomial $f(x)=a_nx^n+...a_0$.
The first is $D(f)=a_n^{2n-2}\prod_{i
The second is $D(f)=(-1)^{\frac{n(n-a)}{2}}\frac{1}{a_n}R(f,f')$ where $R(f,f')$ is the resultant of $f$ and $f'$.