Given a matrix valued rational function $f(\lambda)$ with poles of order $n_k$ at points $\lambda_k$, not at infinity, a book I am following says we can decompose as $$f(\lambda)=f_0+\sum_k f_k(\lambda), \quad f_k(\lambda)=\sum_{r=-n_k}^{-1}f_{k,r}(\lambda-\lambda_k)^r$$ where $f_0$ is a constant.
I seem to think that for a general rational function there would also have to be a piece that is polynomial in $\lambda$ rather than simply a constant term. Is there a way to prove this in general or does it seem like this must just be a definition rather than a general property?
Sources: Ch 3.2 of "Introduction to Classical Integrable Systems" by Babelon, Bernard, and Talon.