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Discriminant

Discriminant
computes the discriminant of the polynomial poly with respect to the variable var.
Discriminant[poly, var, Modulus->p]
computes the discriminant modulo .
  • The discriminant of a polynomial with leading coefficient one is the product over all pairs of roots , of .
  • A Method option can be given, with typical possible values being Automatic, , , , and .
Discriminant of a quadratic:
Discriminant of a quadratic:
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Discriminant of a polynomial with numeric coefficients:
Discriminant of a general cubic:
Discriminant of a general quintic:
Discriminants are squares of differences of roots:
This compares timings of the available methods of discriminant computation:
By default the discriminant is computed over the rational numbers:
Compute the discriminant of the same polynomial over the integers modulo 2:
Compute the discriminant of the same polynomial over the integers modulo 3:
Decide whether a polynomial has multiple roots:
Find the condition for a cubic to have multiple roots:
The discriminant is zero if and only if the polynomial has multiple roots:
The discriminant can be represented in terms of roots as :
Equation relates Discriminant and Resultant:
Using exact coefficients, this indicates no common root:
With approximate coefficients, this does indicate a common root:
in this case, using higher precision resolves the problem:
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