Difference between revisions of "2022 AMC 12A Problems/Problem 21"
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==Problem== | ==Problem== | ||
| − | Let <cmath>P(x) = x^{2022} + x^{1011} + 1</cmath> | + | Let <cmath>P(x) = x^{2022} + x^{1011} + 1.</cmath> Which of the following polynomials is a factor of <math>P(x)</math>? |
<math>\textbf{(A)} \, x^2 -x + 1 \qquad\textbf{(B)} \, x^2 + x + 1 \qquad\textbf{(C)} \, x^4 + 1 \qquad\textbf{(D)} \, x^6 - x^3 + 1 \qquad\textbf{(E)} \, x^6 + x^3 + 1 </math> | <math>\textbf{(A)} \, x^2 -x + 1 \qquad\textbf{(B)} \, x^2 + x + 1 \qquad\textbf{(C)} \, x^4 + 1 \qquad\textbf{(D)} \, x^6 - x^3 + 1 \qquad\textbf{(E)} \, x^6 + x^3 + 1 </math> | ||
Revision as of 13:51, 12 November 2022
Problem
Let ![\[P(x) = x^{2022} + x^{1011} + 1.\]](http://latex.artofproblemsolving.com/c/6/6/c66458c6088164c71caaad852c39637d9cfb6740.png)
Which of the following polynomials is a factor of 
?
Solution
is equal to 
by difference of powers.
Therefore, the answer is a polynomial that divides 
but not 
.
Note that any polynomial 
divides 
if and only if 
is a factor of 
.
The prime factorizations of 
and 
are 
and 
, respectively.
Hence, 
is a divisor of but not .
By difference of powers, 
.
Therefore, the answer is E.
Video Solution by ThePuzzlr
~ MathIsChess
