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Number Theory

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Divisibility in $(\mathbb{Z} / 3\mathbb{Z})[x]$

NUMTH-S95@1O

Recall that $\mathbb{Z} / 3\mathbb{Z} = \{ [a]_3 | 0 \leq a \leq 2, a \in \mathbb{Z} \}$ and $[a]_3 = \{ a + 3b | b \in \mathbb{Z}\}$. Let $x + [1]_{3} \in (\mathbb{Z} / 3\mathbb{Z}) [x]$. Which of the following polynomials are divisible by $x + [1]_{3}$?

Select ALL that apply.

A

$x^2 + [2]_3$

B

$x^4 + x^3 + x + [1]_{3}$

C

$x + [2]_3$

D

$x^2 + [1]_3$