Difference between revisions of "2022 AMC 12A Problems/Problem 21"

Revision as of 01:06, 12 November 2022

$P(x) = x^{2022} + x^{1011} + 1$ 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 E.

@@ Line 8: / Line 8: @@
 The prime factorizations of <math>1011</math> and <math>3033</math> are <math>3*337</math> and <math>3^2*337</math>, respectively.
-Hence, <math>x^9-1</math> is a divisor of <math>x^3033-1</math> but not <math>x^1011-1</math>.
+Hence, <math>x^9-1</math> is a divisor of <math>x^{3033}-1</math> but not <math>x^{1011}-1</math>.
-By difference of powers, <math>x^9-1</math> = <math>(x^3-1)(x^6+x^3+1)</math>.
+By difference of powers, <math>x^9-1=(x^3-1)(x^6+x^3+1)</math>.
 Therefore, the answer is E.