How is the resultant $R(z) =\text{res}_θ(1 - \frac{z}{x}, θ) = 1 - \frac{z}{x}$ calculated?

Question

The resultant of two polynomials is defined as the determinant of the Sylvester matrix. If the polynomials are of degree $n$ and $m$, than the Sylvester matrix will be of dimension

$(m+n)\times (m+n)$.

In the book Algorithms for Computer Algebra on page 534 (example 12.6) the following resultant is calculated:

$R(z) =\text{res}_θ(1 - \frac{z}{x}, θ) = 1 - \frac{z}{x}$

How is this resultant calculated?

Answer 1

I guess the point is to consider $\theta$ as the variable. So the resultant in this case is a $1\times 1$-matrix (since the degree of $1-\frac{z}{x}$ as a polynomial in the indeterminate $\theta$ is $0$ and the degree of $\theta$ is $1$), precisely $[1-\frac{z}{x}]$.