The discriminant $\Delta = 18abcd – 4b^3d + b^2 c^2 – 4ac^3 – 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also that there are three distinct, real roots if $\Delta > 0$, and that there is one real root and two complex roots (complex conjugates) if $\Delta < 0$.
Why does $\Delta < 0$ indicate complex roots? I understand that because of the way that the discriminant is defined, it indicates that there is a repeated root if it vanishes, but why does $\Delta$ greater than $0$ or less than $0$ have special meaning, too?
Best Answer
The discriminant of any monic polynomial is the product $\prod_{i \neq j} (x_i - x_j)^2$ of the squares of the differences of the roots (in an algebraic closure, e.g. $C$). Cf. the Wikipedia article on this. Consequently, if the roots are all real and distinct, this must be positive.
(If the polynomial is not monic, the factor $a_0^{2n-2}$ is thrown in, for $a_0$ the leading coefficient and $n$ the degree; this is positive for a real polynomial.)