I'm self-studying the book of Graphs on Surfaces and Their Applications, Lando & Zvonkin. In Sec 2.3, we assume the form of an underlying polynomial in the Fig.2.20 below as $$ f(x) = K \frac{(x - 1)^3 (x - a)}{x} $$
And to work out the constant $K$, the authors go like
by the fact that the function f − 1 has two double roots...
But I'm wondering like How do we know that it has two double roots?