Cyclic Redundancy Check Codewords Finding the Original Message So I have the received transmitted message in polynomial form as 
$$T'(X) = X^{15}+X^{13}+X^8+X^7+X^4+X^3$$ 
and the error polynomial as 
$$E(X) = X^{16}+X^{13}+X^3$$
 and the generator polynomial as 
$$G(X) = X^4+X^3+X+1.$$ 

I am trying to find $T$ before it is transmitted. It is my understanding that the received transmitted message is the original codeword transmitted ($T$) plus the error polynomial ($E$), but in this case $T'$ is less than $E$, so when I subtract to find $T$, I get a negative polynomial. What am I doing wrong?
You should do arithmetic modulo 2. Modulo 2, $-1$ is the same as $+1$.