# Resultant (algebra)

In algebra, the **resultant** of two polynomials is a quantity which determines whether or not they have a factor in common.

Given polynomials

and

with roots

respectively, the resultant *R*(*f*,*g*) is defined as

The resultant is thus zero if and only if *f* and *g* have a common root.

## [edit] Sylvester matrix

The **Sylvester matrix** attached to *f* and *g* is the square (*m*+*n*)×(*m*+*n*) matrix

in which the coefficients of *f* occupy *m* rows and those of *g* occupy *n* rows.

The determinant of the Sylvester matrix is the resultant of *f* and *g*.

The rows of the Sylvester matrix may be interpreted as the coefficients of the polynomials

and expanding the determinant we see that

*R*(*f*,*g*) =*a*(*x*)*f*(*x*) +*b*(*x*)*g*(*x*)

with *a* and *b* polynomials of degree at most *m*-1 and *n*-1 respectively, and *R* a constant (degree zero polynomial). If *f* and *g* have a polynomial common factor this must divide *R* and so *R* must be zero. Conversely if *R* is zero, then *f*/*g* = - *b*/*a* so *f*/*g* is not in lowest terms and *f* and *g* have a common factor.

## [edit] References

- J.W.S. Cassels (1991).
*Lectures on Elliptic Curves*. Cambridge University Press. ISBN 0-521-42530-1. Chapter 16. - Serge Lang (1993).
*Algebra*, 3rd ed. Addison-Wesley, 200-204. ISBN 0-201-55540-9.

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