example of resultant (1)

To illustrate the concept of resultant, consider a simple example. Let

p(x)=x^{2}-1=(x+1)(x-1)

q(x)=x^{3}-4x=(x+2)x(x-2)

Then, in the notation used in the main entry,

r_{1}=-1\quad r_{2}=+1

s_{1}=-2\quad s_{2}=0\quad s_{3}=+2

Hence,

R(p,q)=(-1-(-2))(-1-0)(-1-2)(1-(-2))(1-0)(1-2)=

\qquad 1\times(-1)\times(-3)\times 3\times 1\times(-1)=-9

In the notation of the main entry,

a_{0}=1\quad a_{1}=0\quad a_{2}=-1

b_{0}=1\quad b_{1}=0\quad b_{2}=-4\quad b_{3}=0

The determinant for computing the resultant is

\left|\begin{matrix}1&0&-1&0&0\cr 0&1&0&-1&0\cr 0&0&1&0&-1\cr 1&0&-4&0&0\cr 0&% 1&0&-4&0\cr\end{matrix}\right|

Since the matrix is quite sparse, its determinant is easy to compute, especially if one first simplifies it by performing some row operations such as subtracting the first row from the fourth row and the second row form the fifth row to make it even sparser. The determinat works out to be $-9$ , in agreement with the earlier answer for the resultant.

Title	example of resultant (1)
Canonical name	ExampleOfResultant1
Date of creation	2013-03-22 14:36:33
Last modified on	2013-03-22 14:36:33
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	5
Author	rspuzio (6075)
Entry type	Example
Classification	msc 13P10