# Sylvester Matrix

Sylvester Matrix

This Demonstration shows the Sylvester matrix of two polynomials and of positive degrees and .

(n+m)×(n+m)

p

q

m

n

The entries of the first row are the coefficients of from highest to lowest power filled out by zeros on the right. The next rows rotate each preceding row to the right, so that the last row ends with the constant term . The last rows are similar, using the coefficients of . See Related Links, "Sylvester Matrix" for a symbolic example.

p

m-1

p(0)

n

q

The determinant of the Sylvester matrix of two polynomials is the resultant of the polynomials (see Related Links).

The polynomials and have a common root if and only if their resultant is zero (see Related Links).

p

q

If , the resultant of and equals [1, pp. 704–707].

p(x)=x-h

p

q

q(h)