# 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

with roots , the discriminant Δ(*f*) is defined as

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

## [edit] Examples

The discriminant of a quadratic *a**X*^{2} + *b**X* + *c* is *b*^{2} − 4*a**c*, which plays a key part in the solution of the quadratic equation.

## [edit] References

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

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