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) 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.

[edit] Examples

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

[edit] References

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