Error Correcting Code understanding

shonen

Why is it that the generating polynomial has a direct effect on the Check digit of the code polynomial. For example say my generating Polynomial was 1 + x + x^2 + x^4 = a0 + a1 (x) + a2 (x^2) + a3(x^3) a4(x^4). Now if a0=0 and a4=0 then the first and last digit of the check digit for the polynomial code would be 0. Lastly, the highest and lowest order coefficient for the generating polynomial is always 1 as a way to conserve check digits.,,what exactly does that mean ?