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

Resultant of polynomials with numeric coefficients:

Resultant of polynomials with parametric coefficients:

Resultant over integers modulo 3:

Resultant over a finite field:

The resultant reflects the multiplicities of roots:

Compute the resultant of two polynomials of degree :

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

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 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 :

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: