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# Resultant

 Resultant[poly1, poly2, var]computes the resultant of the polynomials poly1 and poly2 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 pi-qj 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:
The resultant vanishes exactly when the polynomials have roots in common:
 Out[1]=
 Scope   (4)
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:
 Options   (4)
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:
 Applications   (2)
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:
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