Why every finite language is polynomial? I understand that it's possible to build TM that check all the finite number of cases, so it's definitely in $R$, but I'm not sure why it's in $P$
Suppose that the longest word in the language has length $m$. The Turing machine reads the first $m+1$ symbols on the input tape. Based on that, it can decide whether the input belongs to the language or not. This Turing machine runs in constant time.