Resultant

Remarks

Example: for $f = x - a$ and $g = x - b$, the matrix is $\begin{pmatrix} 1 & -a \\ 1 & -b \end{pmatrix}$, and $\operatorname{Res}(f, g) = a - b$. Let $R = K$ be a field with $a_m \ne 0$ and $b_n \ne 0$, and let $\alpha_1, \dots, \alpha_m$ and $\beta_1, \dots, \beta_n$ be the roots of $f$ and $g$ in a splitting field of $f g$. Then $$\operatorname{Res}(f, g) = a_m^n \, b_n^m \prod_{i=1}^{m} \prod_{j=1}^{n} (\alpha_i - \beta_j).$$ So $\operatorname{Res}(f, g) = 0$ if and only if $f$ and $g$ have a common root, if and only if they have a common factor of positive degree. For instance, $\operatorname{Res}(x^2 - 1, x - 1) = 0$, because of the common root $1$. The value depends on the chosen formal degrees $m$ and $n$: if $a_m = b_n = 0$, the first column is zero and the resultant is $0$ even without a common root. For $f$ of degree $n$ with leading coefficient $a_n$, $\operatorname{Res}(f, f') = (-1)^{n(n-1)/2} a_n \operatorname{Disc}(f)$, where $\operatorname{Disc}$ is the discriminant.