Possible that someone could confirm that I am doing this correctly?
Answer:
The term for $x^8 + x^4 + x^3 + x + 1$ over GF(2) in binary representation is 100011011
representation of $x^8$ is 100000000
$x^8 \bmod x^8 + x^4 + x^3 + x + 1$
(100011011) mod (100000000) = 011011
representation of $x^9$ in binary is = 1000000000
so $x^9\bmod x^8 + x^4 + x^3 + x + 1$
=> (1000000000) mod (100011011) = 010101111
representation of $x^{10}$ in binary is = 10000000000
so $x^{10} \bmod x^8 + x^4 + x^3 + x + 1$
=> (10000000000) mod (100011011) = 01000011
representation of $x^{11}$ in binary is 100000000000
so $x^{11} \bmod x^8 + x^4 + x^3 + x + 1$
=> (100000000000) mod (100011011) = 010000110
representation of $x^{12}$ in binary is 1000000000000
so $x^{12} \bmod x^8 + x^4 + x^3 + x + 1$
=> (1000000000000) mod (100011011) = 0100001100
representation of $x^{13}$ in binary is 10000000000000
so $x^{13} mod x^8 + x^4 + x^3 + x + 1$
=> (10000000000000) mod (100011011) = 011111101
representation of $x^{14}$ binary is 100000000000000
so $x^{14} \bmod x^8 + x^4 + x^3 + x + 1$
=> (100000000000000) mod (100011011) = 011111101
representation of $x^{15}$ in binary is 1000000000000000
so $x^{15} mod x^8 + x^4 + x^3 + x + 1$
=> (1000000000000000) mod (100011011) = 011011111