# Discriminant of a polynomial

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

Given a polynomial

$f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

with roots $\alpha_1,\ldots,\alpha_n$, the discriminant Δ(f) is defined as

$\Delta = (-1)^{n(n-1)/2} a_n^{2(n-1)} \prod_{i \neq j} (\alpha_i - \alpha_j) .$

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

In spite of the definition in terms of the roots, Δ(f) appears to be a polynomial function of the coefficients $a_1,\ldots,a_n$ and may be obtained as the resultant of the polynomial and its formal derivative.

## Examples

The discriminant of a quadratic aX2 + bX + c is b2 − 4ac, which plays a key part in the solution of the quadratic equation.

## References

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