Resultant

Given a Polynomial $p(x)$ of degree $n$ with roots $\alpha_i$, $i=1$, ..., $n$ and a Polynomial $q(x)$ of degree $m$ with roots $\beta_j$, $j=1$, ..., $m$, the resultant is defined by

$$R(p,q)=\prod_{i=1}^n \prod_{j=1}^m (\beta_j-\alpha_i).$$

There exists an Algorithm similar to the Euclidean Algorithm for computing resultants (Pohst and Zassenhaus 1989). The resultant is the Determinant of the corresponding Sylvester Matrix. Given $p$ and $q$, then

$$h(x)=R(q(t),p(x-t))$$

is a Polynomial of degree $mn$, having as its roots all sums of the form $\alpha_i+\beta_j$.

See also Discriminant (Polynomial), Sylvester Matrix

References

Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Number Theory. Cambridge, England: Cambridge University Press, 1989.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, p. 348, 1991.