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

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

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