Here is the question I'm currently looking at:
Show that the discriminant of the equation $y^3+py+q=0$ is $-4p^3-27q^2$.
I've done some research and found this, but we haven't studied Vieta's theorem in my math class just yet. All I know is that if we take a cubic equation of the form $$x^3+ax^2+bx+c=0$$
and we manipulate it to get rid of the $x^2$ term, then we yield $$y^3+py+q=0$$
where $$p=\frac{3b-a^2}{3}$$ and $$q=\frac{27c+2a^3-9ab}{27}$$
Other than that, I don't know how to approach this question.
Best Answer
Any $y^3 + py + q$ can be written as $F := (y-x_1)(y-x_2)(y+x_1 + x_2)$ for some $x_1,x_2$ in an algebraic extension. So it suffices to prove the formula just for this one polynomial $F$. Compute the discriminant of $F$ directly from the definition, compute the $x^1$ and $x^0$ coefficients in $F$ (i.e. $p$ and $q$) in terms of $x_1,x_2$, then compute $-4p^3-27q^2$ and compare.