I read it in the book Rational Points on Elliptic Curves by Silverman and Tate:
If $f(x)$ is a polynomial with leading coefficient 1 in $Z[x]$, then the discriminant of $f(x)$ will be in the ideal $<f(x), f'(x)>$
It says it follows from general theory of discriminant, but what's general theory? Or how to prove it instead?
The discriminant of a polynomial $f$ of degree $n$ and leading coefficient $a_n$ is $$\operatorname{disc}f=\frac{(-1)^{\tfrac{n(n-1)}2}}{a_n}\operatorname{Res}(f,f').$$ ($\operatorname{Res}(f,g)$ denotes the resultant of the polynomials $f$ and $g$). Thus it is $0$ if and only if $f$ has a multiple root.
For any polynomials $f,g$, it can be shown there exist polynomials $u,v$ such that $$\operatorname{Res}(f,g)=uf+vg.$$ Hence the assertion for the discriminant of $f$.