Definition:Discriminant of Polynomial

Definition
Let $f \left({x}\right)$ be a polynomial in $x$ of degree $n$ over a field $k$.

That is, let $f \left({x}\right) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1} x + a_n$ with $a_i \in k$, $i = 1,\ldots, n$.

Let the roots of the equation $f \left({x}\right) = 0$ be $\alpha_1, \alpha_2, \ldots, \alpha_n$.

Then the discriminant $\Delta(f)$ of $f$ is defined as:


 * $\displaystyle \Delta \left({f}\right) := \prod_{1 \le i < j \le n} \left({\alpha_i - \alpha_j}\right)^2$.

Quadratic Equation
The concept is usually encountered in the context of a quadratic equation $a x^2 + b x + c$:

Cubic Equation
In the context of a cubic equation $a x^3 + b x^2 + c x + d$: