Final answer:
To check if the received CRC code is legal with a key, divide the received CRC code by the key using polynomial long division. If the remainder is zero, the code is legal. In this case, the received CRC code is not legal with the given key.
Step-by-step explanation:
To check if the received CRC code is legal with a key, we need to perform the CRC division. Let's represent the received CRC code as a polynomial, C(x), and the key as K(x). Then we divide C(x) by K(x) using polynomial long division. If the remainder is zero, then the received CRC code is legal. In this case, the received CRC code is 1000110 and the key is 1011.
Representing 1000110 as a polynomial, C(x), we have:
C(x) = x^6 + x^5 + x
Representing 1011 as a polynomial, K(x), we have:
K(x) = x^3 + 1
Now we perform the polynomial long division:
______________________
K(x) | x^6 + x^5 + x
- (x^6 + + x)
-----------------
x^5 + x
We have a remainder of x^5 + x. Since the remainder is not zero, the received CRC code is not legal with the given key.