# Resultant

Resultant[poly1,poly2,var]

computes the resultant of the polynomials poly1 and poly2 with respect to the variable var.

Resultant[poly1,poly2,var,Modulusp]

computes the resultant modulo the prime p.

# Details and Options

• The resultant of two polynomials p and q, both with leading coefficient 1, is the product of all the differences pi-qj between roots of the polynomials. The resultant is always a number or a polynomial.

# Examples

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## Basic Examples(1)

The resultant vanishes exactly when the polynomials have roots in common:

## Scope(5)

Resultant of polynomials with numeric coefficients:

Resultant of polynomials with parametric coefficients:

Resultant over integers modulo 3:

The resultant reflects the multiplicities of roots:

Compute the resultant of two polynomials of degree :

## Generalizations & Extensions(1)

The resultant of rational functions is defined using the multiplicative property:

## Options(4)

### Method(1)

This compares timings of the available methods of resultant computation:

### Modulus(3)

By default the resultant is computed over the rational numbers:

Compute the resultant of the same polynomials over the integers modulo 2:

Compute the resultant of the same polynomials over the integers modulo 3:

## Applications(2)

Decide whether two polynomials have common roots:

Find conditions for two polynomials to have common roots:

## Properties & Relations(6)

The resultant is zero if and only if the polynomials have a common root:

The polynomials have a zero resultant if and only if they have a nonconstant PolynomialGCD:

The resultant can be represented in terms of roots as :

Equation relates Discriminant and Resultant:

GroebnerBasis can also be used to find conditions for common roots:

The same problem can also be solved using Reduce, Resolve, and Eliminate:

## Possible Issues(1)

The following two polynomials have no common root:

Using approximate coefficients they will appear to have a common root:

Using higher precision shows they have no common root:

Wolfram Research (1988), Resultant, Wolfram Language function, https://reference.wolfram.com/language/ref/Resultant.html (updated 2022).

#### Text

Wolfram Research (1988), Resultant, Wolfram Language function, https://reference.wolfram.com/language/ref/Resultant.html (updated 2022).

#### CMS

Wolfram Language. 1988. "Resultant." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Resultant.html.

#### APA

Wolfram Language. (1988). Resultant. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Resultant.html

#### BibTeX

@misc{reference.wolfram_2022_resultant, author="Wolfram Research", title="{Resultant}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Resultant.html}", note=[Accessed: 20-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_resultant, organization={Wolfram Research}, title={Resultant}, year={2022}, url={https://reference.wolfram.com/language/ref/Resultant.html}, note=[Accessed: 20-March-2023 ]}