The discriminant for cubic equations is -
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 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.