Let Q(z) and R(z) be the unique polynomials such that &z^2021+1 & = (z^2+z+1)Q(z)+R(z) and the degree of R is less than 2. What is R(z)?

2021 AMC 12B Spring Problem 20

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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

Modulo

z^2+z+1

we have

z^3\equiv 1,

since

z^3-1=(z-1)(z^2+z+1)

Reduce the exponent

2021

modulo

3,

then reduce

z^2

using

z^2\equiv -z-1

Since

z^3\equiv 1\pmod{z^2+z+1}

and

2021=3\cdot 673+2,

we have

z^{2021}\equiv z^2.

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.