# 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) ExampleOfResultant1 2013-03-22 14:36:33 2013-03-22 14:36:33 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Example msc 13P10