BUILT-IN MATHEMATICA SYMBOL

Resultant

computes the resultant of the polynomials

and

with respect to the variable var.

Resultant[poly₁, poly₂, var, Modulus->p]
computes the resultant modulo the prime p.

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

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

Out[1]=

Resultant of polynomials with numeric coefficients:

Resultant of polynomials with parametric coefficients:

Resultant over integers modulo 3:

The resultant reflects the multiplicities of roots:

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

This compares timings of the available methods of resultant computation:

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:

Decide whether two polynomials have common roots:

Find conditions for two polynomials to have common roots:

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

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

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

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: