The discriminant for cubic equations is –

$Δ:=b^22:^2−4ac^3−4b^3d−27a^2d^2+18abcd$

And I am aware that you can determine the number of roots a cubic has using method shown below –

$Δ:>0$ the equation has three distinct real roots

$Δ:=0$ the equation has a repeated root and all its roots are real

$Δ:<0$ the equation has one real root and two non-real complex conjugate roots

But I was wondering if one could determine whether a cubic has rational or integer roots, as you can do with the discriminant for quadratics, and if so what the method would be.

I have noticed that with the cubics I have checked: if the discriminant is a perfect square there are 3 integer solutions, although I have not checked many and I am not sure of the reasoning behind it.

Any help would be greatly appreciated.

When a monic cubic has square discriminant but no rational roots, what we expect is real roots that can be written as (doubled) cosines, or sums of them.

$$ x^3 + x^2 – 2x – 1 $$

has $$ 2 cos frac{2 pi}{7} ; , ; ; 2 cos frac{4 pi}{7} ; , ; ; 2 cos frac{8 pi}{7} ; , ; ; $$

more in a minute

$$ x^3 – 3x + 1 $$

has $$ 2 cos frac{2 pi}{9} ; , ; ; 2 cos frac{4 pi}{9} ; , ; ; 2 cos frac{8 pi}{9} ; , ; ; $$

$$ $$

$$ x^3 + x^2 – 4x + 1 $$

has $$ 2 cos frac{2 pi}{13} + 2 cos frac{10 pi}{13}; , ; ; 2 cos frac{4 pi}{13} + 2 cos frac{6 pi}{13} ; , ; ; 2 cos frac{8 pi}{13} +2 cos frac{12 pi}{13} ; , ; ; $$