2021 AMC 12B Spring Problem 20

Attempt Problem 20 of the 2021 AMC 12B Spring below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

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20.

Let $Q(z)$ and $R(z)$ be the unique polynomials such that $\begin{aligned} &z^{2021}+1 \\ &\quad = (z^2+z+1)Q(z)+R(z) \end{aligned}$ and the degree of $R$ is less than $2.$ What is $R(z)?$

$-z$

$-1$

$2021$

$z+1$

$2z+1$

Answer: A

Concepts:polynomial roots of unity modular arithmetic

Difficulty rating: 1990

Solution:

Since $z^3\equiv 1\pmod{z^2+z+1}$ and $2021=3\cdot 673+2,$ we have $z^{2021}\equiv z^2.$

So $z^{2021}+1\equiv z^2+1.$ Reducing further with $z^2\equiv -z-1,$ this is $-z-1+1=-z.$

Therefore $R(z)=-z.$

Thus, the correct answer is A.

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