we have fixed polynomial $f(x)=x^3+bx^2+cx+d$ I need to proof that polynomial`s roots $\alpha,\beta ,\gamma$ are Real if and only if the discriminant
$D(f)= (-1)^{\binom{3}{2}}*Res(p,p`)= b^2c^2-4c^3-4b^3d-27d^2+18bcd \ge 0 $
I tried the first side of the proof by assuming that $\alpha,\beta,\gamma$ are real. we can write the polynomial $f(x)=(x-\alpha)(x-\beta)(x-\gamma)$ by opening the brackets we get that $b=-(\alpha+\beta+\gamma) , c=(\gamma(\alpha+\beta)+\alpha\beta) ,d=-\alpha\beta\gamma$ I assume by contradiction that $D(f) \lt 0 $ but i only get the inequallity $[(\alpha+\beta+\gamma)(\gamma(\alpha+\beta)+\alpha\beta)]*[(\alpha+\beta+\gamma)(\gamma(\alpha+\beta)+\alpha\beta)+18\alpha\beta\gamma]+4(\alpha+\beta+\gamma)\lt 4(\gamma(\alpha+\beta)+\alpha\beta)^3+27\alpha^2\beta^2\gamma^2$ I don`t know what to do from here please help.