Sylvester Matrix

m

1

2

3

n

1

2

3

4

new polynomials

grid/matrix

show determinant

p(x)x-1

q(x)2x-2

A = 

1	-1
2	-2



This Demonstration shows the

(n+m)×(n+m)

Sylvester matrix of two polynomials

p

and

q

of positive degrees

m

and

n

.

The entries of the first row are the coefficients of

p

from highest to lowest power filled out by zeros on the right. The next

m-1

rows rotate each preceding row to the right, so that the last row ends with the constant term

p(0)

. The last

n

rows are similar, using the coefficients of

q

. See Related Links, "Sylvester Matrix" for a symbolic example.

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

The polynomials

p

and

q

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

If

p(x)=x-h

, the resultant of

p

and

q

equals

q(h)

[1, pp. 704–707].