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 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