Interior zero count for a finite Blaschke product minus a constant ($N(B-a)=\mathrm{deg}B(|a|<1)$)

A finite Blaschke product is holomorphic across the unit circle and has modulus one there. For a constant $a$ with $|a|<1$, Rouché's theorem makes $B-a$ and $B$ have the same number of interior zeros, counting multiplicity. Each factor contributes its prescribed zero; the poles are outside the closed disk.