Recall that a generator polynomial \( g(x) \) of degree \( r \) can detect all single-bit errors if it has at least two terms (i.e., it is not of the form \( x^k \)).

Consider how the remainder \( r(x) \) behaves when a single-bit error occurs in the transmitted codeword \( c(x) \). What must be true about \( r(x) \) for the error to be undetectable?

Prove that if \( g(x) \) has at least two terms, then \( g(x) \) does not divide \( x^i + 1 \) for any \( i \). How does this ensure single-bit errors are detected?