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Discriminant[poly,var]

computes the discriminant of the polynomial poly with respect to the variable var.

Discriminant[poly,var,Modulusp]

computes the discriminant modulo .

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Modulus  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
Cite this Page

Discriminant

Discriminant[poly,var]

computes the discriminant of the polynomial poly with respect to the variable var.

Discriminant[poly,var,Modulusp]

computes the discriminant modulo .

Details and Options

  • The discriminant of a polynomial with leading coefficient one is the product over all pairs of roots , of .
  • Discriminant takes the following options:
  • Method Automaticmethod to use
    Modulus 0modulus to assume for integers
  • Typical possible values for Method are Automatic, "SylvesterMatrix", "BezoutMatrix", "Subresultants" and "Modular".

Examples

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Basic Examples  (1)

Discriminant of a quadratic:

Wolfram Language code: Discriminant[a x ^ 2 + b x + c, x]

Scope  (7)

Discriminant of a polynomial with numeric coefficients:

Wolfram Language code: Discriminant[x ^ 10 - 5x ^ 7 - 3 x + 9, x]

Discriminant of a general cubic:

Wolfram Language code: Discriminant[a x ^ 3 + b x ^ 2 + c x + d, x]

Discriminant of a general quintic:

Wolfram Language code: Discriminant[Sum[Subscript[a, i]x ^ i, {i, 0, 5}], x]

Discriminants are squares of differences of roots:

Wolfram Language code: Discriminant[(x - a)(x - b)(x - c)(x - d), x]

Discriminant over integers modulo 3:

Wolfram Language code: Discriminant[x ^ 5 - x y + y ^ 2 - 1, x, Modulus -> 3]

Discriminant over a finite field:

Wolfram Language code: ℱ = FiniteField[17, 3];
Wolfram Language code: Discriminant[ℱ[1]x ^ 5 + ℱ[123]x y + ℱ[456], x]

Compute the discriminant of a polynomial of degree :

Wolfram Language code: rpoly[n_] := RandomInteger[{-2 ^ 10, 2 ^ 10}, {n + 1}].x ^ Range[0, n] SeedRandom[1234]; p = rpoly[1000];
Wolfram Language code: Discriminant[p, x]//Short//AbsoluteTiming

Options  (4)

Method  (1)

This compares timings of the available methods of discriminant computation:

Wolfram Language code: Timing[Discriminant[x ^ 100 - 2x ^ 77 + 3x + 4, x, Method -> #]//Short]& /@ {Automatic, "Modular", "Subresultants", "BezoutMatrix", "SylvesterMatrix"}//Column
Wolfram Language code: Timing[Discriminant[a x ^ 10 + b x ^ 5 + (a + b)x + a b c, x, Method -> #]//Short]& /@ {Automatic, "Modular", "Subresultants", "BezoutMatrix", "SylvesterMatrix"}//Column

Modulus  (3)

By default the discriminant is computed over the rational numbers:

Wolfram Language code: Discriminant[a x ^ 3 + b x ^ 2 + c x + d, x]

Compute the discriminant of the same polynomial over the integers modulo 2:

Wolfram Language code: Discriminant[a x ^ 3 + b x ^ 2 + c x + d, x, Modulus -> 2]

Compute the discriminant of the same polynomial over the integers modulo 3:

Wolfram Language code: Discriminant[a x ^ 3 + b x ^ 2 + c x + d, x, Modulus -> 3]

Applications  (2)

Decide whether a polynomial has multiple roots:

Wolfram Language code: Discriminant[x ^ 11 - 2x ^ 10 + x ^ 9 - 2x ^ 3 + 11x ^ 2 - 16x + 7, x]
Wolfram Language code: FactorSquareFree[x ^ 11 - 2x ^ 10 + x ^ 9 - 2x ^ 3 + 11x ^ 2 - 16x + 7]
Wolfram Language code: Discriminant[x ^ 11 - 2x ^ 10 + x ^ 9 - 2x ^ 3 + 11x ^ 2 - 16x - 7, x]
Wolfram Language code: FactorSquareFree[x ^ 11 - 2x ^ 10 + x ^ 9 - 2x ^ 3 + 11x ^ 2 - 16x - 7]

Find the condition for a cubic to have multiple roots:

Wolfram Language code: Solve[Discriminant[x ^ 3 + x + c , x] == 0, c]
Wolfram Language code: FactorSquareFree[x ^ 3 + x + c /. %, Extension -> Automatic]

Properties & Relations  (3)

The discriminant is zero if and only if the polynomial has multiple roots:

Wolfram Language code: Discriminant[(x - 1)(x - 2)(x - 3), x]
Wolfram Language code: Discriminant[(x - 1)(x - 2)(x - 1), x]

The discriminant can be represented in terms of roots as :

Wolfram Language code: x /. Solve[a x ^ 2 + b x + c == 0, x]
Wolfram Language code: Expand[a ^ (2 2 - 2)(%[[1]] - %[[2]]) ^ 2]
Wolfram Language code: Discriminant[a x ^ 2 + b x + c, x]

Equation relates Discriminant and Resultant:

Wolfram Language code: f = 3x ^ 7 - 5x + 4;
Wolfram Language code: Resultant[f, D[f, x], x]
Wolfram Language code: (-1) ^ (7 6 / 2)Coefficient[f, x, 7]Discriminant[f, x]

Possible Issues  (1)

Using exact coefficients, this indicates no common root:

Wolfram Language code: Discriminant[(x - 1 + 10 ^ (-17))(x - 1), x]

With approximate coefficients, this does indicate a common root:

Wolfram Language code: Discriminant[(x - 1. + 10 ^ (-17))(x - 1), x]

in this case, using higher precision resolves the problem:

Wolfram Language code: Discriminant[(x - 1`20 + 10 ^ (-17))(x - 1), x]
Wolfram Research (2007), Discriminant, Wolfram Language function, https://reference.wolfram.com/language/ref/Discriminant.html (updated 2023).

Text

Wolfram Research (2007), Discriminant, Wolfram Language function, https://reference.wolfram.com/language/ref/Discriminant.html (updated 2023).

CMS

Wolfram Language. 2007. "Discriminant." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/Discriminant.html.

APA

Wolfram Language. (2007). Discriminant. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Discriminant.html

BibTeX

@misc{reference.wolfram_2026_discriminant, author="Wolfram Research", title="{Discriminant}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/Discriminant.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_discriminant, organization={Wolfram Research}, title={Discriminant}, year={2023}, url={https://reference.wolfram.com/language/ref/Discriminant.html}, note=[Accessed: 13-June-2026]}