# Discriminant of a polynomial

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In algebra, the discriminant of a polynomial is an invariant which determines whether or not a polynomial has repeated roots.

Given a polynomial

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$

with roots ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$, the discriminant Δ(f) with respect to the variable x is defined as

${\displaystyle \Delta =(-1)^{n(n-1)/2}a_{n}^{2(n-1)}\prod _{i\neq j}(\alpha _{i}-\alpha _{j}).}$

The discriminant is thus zero if and only if f has a repeated root.

The discriminant may be obtained as the resultant of the polynomial and its formal derivative.

## Examples

The discriminant of a quadratic ${\displaystyle aX^{2}+bX+c}$ is ${\displaystyle b^{2}-4ac}$, which plays a key part in the solution of the quadratic equation.