How this calculator works
The discriminant vanishes when a polynomial shares a root with its derivative. This tool constructs their Sylvester matrix and evaluates its determinant, then adjusts the sign and leading coefficient. For quadratics and cubics, the sign determines the familiar real-root cases. For higher degrees it carries less information. The determinant is numerical, so nearly repeated roots may need exact symbolic arithmetic.
Formula
Disc(P) = (-1)^(n(n-1)/2) Res(P,P') / a_n. For degree 2, Disc = b^2 - 4ac.
Worked example
For x^2 - 5x + 6, the discriminant is 25 - 24 = 1. For x^3 - 1 it is -27, consistent with one real root and two non-real roots. For x^4 - 1 it is -256.
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Frequently asked questions
Does a negative result always mean no real roots?
Only for a quadratic. A real cubic with negative discriminant still has one real root, and higher degrees need more analysis.
What does zero mean?
At least one complex root is repeated. Numerical cancellation can affect whether a computed result is exactly zero.
Why enter the degree?
The chosen degree fixes the coefficient count and prevents accidentally treating a lower-degree polynomial as a higher-degree one.