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) with respect to the variable x 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.

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

Examples

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

References

• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 193-194,204-205,325-326. ISBN 0-201-55540-9.