To find the remainder when dividing a polynomial, we can use polynomial long division. In this case, we need to divide x³ + 1 by x² + x + 1.
1. First, we set up the long division. We divide the leading term of x³ (which is x³) by the leading term of x² (which is x²) to get x. We then multiply x by the entire divisor:
x * (x² + x + 1) = x³ + x² + x
2. Next, we subtract this result from the original polynomial:
(x³ + 1) – (x³ + x² + x) = -x² – x + 1
3. Now we repeat the process. We divide -x² by x² to get -1. We multiply -1 by the divisor:
-1 * (x² + x + 1) = -x² – x – 1
4. We subtract this result from -x² – x + 1:
(-x² – x + 1) – (-x² – x – 1) = 2
This final result, 2, is the remainder. So, when we divide x³ + 1 by x² + x + 1, the remainder is:
The answer is: 2