2022 AMC 12A Problems/Problem 21
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 
.
Solution 2
We simply test roots for each, as 
are multiples of three, we need to make sure the roots are in the form of $e^{i\frac{k\pi}{9}$ (Error compiling LaTeX. Unknown error_msg), so we only have to look at 
.
If we look at choice 
, $x=e^{i\frac{\pm2\pi}{9}$ (Error compiling LaTeX. Unknown error_msg) which works perfectly, the answer is just
~bluesoul
Video Solution by ThePuzzlr
~ MathIsChess
See Also
| 2022 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 20 |
Followed by Problem 22 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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