AMC12 2022 A

AMC12 2022 A · Q21

AMC12 2022 A · Q21. It mainly tests Primes & prime factorization, GCD & LCM.

Let \[P(x) = x^{2022} + x^{1011} + 1.\] Which of the following polynomials is a factor of $P(x)$?

设 \[P(x) = x^{2022} + x^{1011} + 1.\] 以下哪个多项式是 $P(x)$ 的因式？

(A) \, x^2 -x + 1 \, x^2 -x + 1

(B) \, x^2 + x + 1 \, x^2 + x + 1

(D) \, x^6 - x^3 + 1 \, x^6 - x^3 + 1

(E) \, x^6 + x^3 + 1 \, x^6 + x^3 + 1

Answer

Correct choice: (E)

正确答案：（E）

Solution

$P(x) = x^{2022} + x^{1011} + 1$ is equal to $\frac{x^{3033}-1}{x^{1011}-1}$ by difference of powers. Therefore, the answer is a polynomial that divides $x^{3033}-1$ but not $x^{1011}-1$. Note that any polynomial $x^m-1$ divides $x^n-1$ if and only if $m$ is a factor of $n$. The prime factorizations of $1011$ and $3033$ are $3*337$ and $3^2*337$, respectively. Hence, $x^9-1$ 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 $\boxed{E}$.

$P(x) = x^{2022} + x^{1011} + 1$ 等于 $\frac{x^{3033}-1}{x^{1011}-1$，由差幂公式得。因此，答案是除 $x^{3033}-1$ 但不除 $x^{1011}-1$ 的多项式。注意，多项式 $x^m-1$ 除 $x^n-1$ 当且仅当 $m$ 是 $n$ 的因数。 $1011$ 和 $3033$ 的质因数分解分别是 $3\times337$ 和 $3^2\times337$。因此，$x^9-1$ 是 $x^{3033}-1$ 的因式但不是 $x^{1011}-1$ 的。由差幂公式，$x^9-1=(x^3-1)(x^6+x^3+1)$。因此，答案是 $\boxed{E}$。

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