CRC generator polynomial

Hi, I have a question on the basics of choosing a CRC generator polynomial. Books mention that if the polynomial has "1+x" as a factor it implies that the corrosponding codeword will have even number of terms. Furthermore the parity is even and any number of odd errors can be detected.

If a poly has "x" as a factor it cna detect all double errors.

Any explation why this is so ?

regards ashu

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