Detecting errors changing an odd number of bits using CRC

Question

I was studying CRC from lecture notes of Schwarzkopf but I am stuck at this statement:

If $G(x)$ is a factor of $E(x)$, then $G(1)$ would also have to be $1$.

What does this statement means? It is the third point in the third last block of the article whose link is given. I know a similar question is asked but that is a specific question. I want to understand this particular statement.

Answer 1

Recall that we are working modulo $2$. Thus $E(x)$ is a polynomial whose coefficients are $0,1$, and $E(1) \in \{0,1\}$.

By definition, $G(x)$ is a factor of $E(x)$ if there exists a polynomial $H(x)$ such that $E(x) = G(x) H(x)$.

In this case, we assume that $E(1) = 1$. Since $E(1) = G(1) H(1)$, this forces $G(1) = 1$.