Resultant (algebra)

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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) with respect to the variable x is defined as

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

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

with a and b polynomials of degree at most m-1 and n-1 respectively, and R a scalar. 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.

References