Given

• Message (dataword): 11001001
• Generator polynomial: $x^3 + 1 \Rightarrow$ divisor 1001 (degree 3, so append 3 zeros)

Step 1: Append zeros

11001001 + 000 = 11001001000

Step 2: Modulo-2 division by 1001

Step	Current bits	XOR with	Remainder
1	1100	1001	101
2	101+1 = 1011	1001	010
3	010+0 = 0100	0000	100
4	100+0 = 1000	1001	001
5	001+1 = 0011	0000	011
6	011+0 = 0110	0000	110
7	110+0 = 1100	1001	101
8	101+0 = 1010	1001	011

CRC (remainder) = 011

Check using polynomials: since $x^3 \equiv 1 \pmod{x^3+1}$, the message $x^7+x^6+x^3+1$ times $x^3$ gives $x^{10}+x^9+x^6+x^3 \equiv x + 1 + 1 + 1 = x + 1$, i.e. 011. ✔

Step 3: Transmitted message

Dataword + CRC = 11001001 | 011  ->  11001001011

Receiver check

Dividing 11001001011 by 1001 gives remainder 000, so no error is detected.