how to prove the periodity of an LFSR
Hint: The LFSR changes its state by multiplying it by $x$ modulo $c(x)$. So if the initial state is $x_0$, the state after $t$ steps is $x_t \equiv x_0 x^t \pmod{c(x)}$. In particular, $x_t = x_0$ if $x_0 (x^t-1) \equiv 0 \pmod{c(x)}$.