2022 AMC 12A Problems/Problem 21
Problem
Let 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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