As per Wikipedia, the discriminant of a polynomial $f(x)=a_nx^n+$(lower degree terms) is defined in terms of roots of $f(x)$ as $$ D(f)=a_n^{2n-2}\prod_{i<j} (\alpha_i-\alpha_j)^2. $$ My question is simple, but I am not able to find answer: why the coefficient $a_n$ is taken with power $2n-2$ in the definition of discriminant? (Lang in his Algebra mentions same expression for $D(f)$, p.204.)
To ask more precisely, the above quantity depends on the roots of $f(x)$ (and independent of ordering of roots); since a monic polynomial $\tilde{f}$ and $a_n\tilde{f}$ (with $a_n\neq 0$) have same roots, one may ignore $a_n$ in defining discriminant. But, if one wants to consider leading coefficient in defining discriminant, why it is taken, as above, with power $2n-2$? Not $n$ or $2n$?