= Resultant as a product over roots
{title2=$\operatorname{Res}(f,g)=a_m^n\prod_{i=1}^m g(\alpha_i)$}
Write $f(x)=a_m\prod_{i=1}^m(x-\alpha_i)$ and $g(x)=b_n\prod_{j=1}^n(x-\beta_j)$ in an <algebraic closure>, counting <multiplicity of a root>. The coefficient-row convention above gives
$$
\operatorname{Res}(f,g)
=a_m^n\prod_{i=1}^m g(\alpha_i)
=a_m^n b_n^m\prod_{i=1}^m\prod_{j=1}^n(\alpha_i-\beta_j).
$$
This makes the shared-<root of a polynomial> test transparent: a factor vanishes precisely when one $\alpha_i$ equals one $\beta_j$. Swapping the two <polynomials> reverses the sign of all $mn$ differences, giving $\operatorname{Res}(g,f)=(-1)^{mn}\operatorname{Res}(f,g)$.<Algebraic Techniques and Semidefinite Optimization, MIT 6.972, Spring 2006/Lecture 6: Resultants and Discriminants>[Algebraic Techniques and Semidefinite Optimization, MIT 6.972, Spring 2006; MIT 6.972, Lecture 6, pp. 6-1–6-2, root products, equation (1), and argument-swap sign]{ref}
The value measures an algebraic product, not a normalized geometric distance. Replacing $f$ by $cf$ for nonzero $c\in K$ leaves its <roots of a polynomial> unchanged but multiplies the <resultant> by $c^n$. Therefore a small numerical <resultant> alone does not quantify how close two sets of <roots of a polynomial> are. Exact zero testing and numerical sensitivity are different questions.
A <multiple root> of a <polynomial> is also a <root of a polynomial> of its <formal derivative>. In <characteristic zero>, for $\deg f=m\geq2$, the associated <discriminant> is
$$
\operatorname{Disc}(f)=\frac{(-1)^{m(m-1)/2}}{a_m}\operatorname{Res}(f,f').
$$
For example, $h(x)=x^2+s$ gives $\operatorname{Res}(h,2x)=4s$ and $\operatorname{Disc}(h)=-4s$. The two <roots of a polynomial> merge precisely at $s=0$.<Algebraic Techniques and Semidefinite Optimization, MIT 6.972, Spring 2006/Lecture 6: Resultants and Discriminants>[Algebraic Techniques and Semidefinite Optimization, MIT 6.972, Spring 2006; MIT 6.972, Lecture 6, p. 6-2, Definition 2: discriminant as a polynomial–derivative resultant]{ref}
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