<s>[INST] Characterize the elements of a set of polynomials Given $$M = \{x^3 + ax^2 + bx + c \in\mathbb Z_3[x]\}$$ and given $$A=\{f \in M \mid\overline{1} \text{ is a root of }\ f\}.$$
The exercise asks to characterize (tell) what are the elements of $A$ and how many of them there are. My first attempt at solving this exercise was by using Ruffini's rule: 
I divided $$x^3 + ax^2 + bx + c$$ by $$x-1$$ and the result is $$(x^2+\overline2 ax + \overline2 x + b)(x-\overline1) + c \in\mathbb Z_3[x]$$Is this a correct solution for this exercise?
 [/INST] You want all polynomials $x^3+ax^2+bx+c$ such that $1+a+b+c=0\bmod3\ \ (*)$. Pick any $a$ and any $b$, then $(*)$ holds iff $c=2-a-b\bmod3$. So we have a total of 9 solutions. If you want them explicitly:
$x^3+2,x^3+x+1,x^3+2x$,
$x^3+x^2+1,x^3+x^2+x,x^3+x^2+2x+2$,
$x^3+2x^2,x^3+2x^2+x+2,x^3+2x^2+2x+1$.
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