Let $f(x)$ be a non-constant polynomial of degree $n$ over a field $F$ and $r_1,r_2,\ldots, r_n$ be its roots in some extension of $F$.
The discriminant of $f(x)$ is a symmetric expression in its roots, but, a little modification comes when the polynomial is non-monic.
(i) If $f(x)$ is monic, then discriminant is defined as $\prod_{i<j} (r_i-r_j)^2$.
(ii) If $f(x)$ has leading term $a_n(\neq 0)$, then its discriminant is defined as $a_n^{2n-2} \prod_{i<j} (r_i-r_j)^2$.
Question Why the modification for leading term $a_n$ in the definition of discriminant is done in above manner? (I mean, why discriminant is not defined simply by the expression of roots as above by ignoring leading term?) If we see further the connection of discriminant with resultant, the leading term still doesn't miss even if we include $a_n^{2n-2}$ in the discriminant expression since, by (ii), the following is connection with discriminant: $$ \mbox{discriminant}(f)= \frac{(-1)^{n(n-1)/2}\mbox{Resultant}(f,f')}{a_n}. $$