2022 AMC 12A Problems/Problem 21

Solution

P(x) = x^2022 + x^1011 + 1 = (x^3033 - 1) / (x^1011 - 1) by difference of powers Therefore, the answer is a polynomial that divides x^3033 - 1 but not x^1011 - 1.

Note any polynomial (x^m - 1) divides (x^n - 1) if and only if m is a factor of n. 1011 = 3*337, 3033 = 3^2 * 337 => x^9 - 1 is a divisor of x^3033 - 1 but not x^1011 - 1.

By difference of powers, x^9 - 1 = (x^3 - 1)(x^6 + x^3 + 1) Therefore, the answer is E.