When Errors Go Undetected in CRC

Question

I understand that CRC will not be able to detect errors if:

1. The remainder of $E(x) / G(x) = 0$
2. $E(x) = G(x).Z(x)$ for some polynomial $Z(x)$

I understand the first point, which means that if the error polynomial is divisible by the generator without remainder, then you can't detect the error. This is because the received message $C'(x) = C(x) + E(x)$. So if $E(x) / G(x)$, then $C(x) / G(x)$ and $C'(x)/G(x)$.

But I don't understand the second point. What is $Z(x)$ in the context of a CRC? Why would it mean a CRC could not detect an error if $G(x).Z(x)$ was equal to the error polynomial?

Answer 1

The second case reduces to the first, because if $E(x) = G(x).Z(x)\;$ the remainder $R(x)$ in the division $E(x) / G(x)$ is zero, assuming your operations . and \ are defined by $E(x) = G(x).Z(x) +R(x),\;$ i.e. $Z$ is the quotient and $R$ is the remainder.