# Elements of quotient ring $\mathbb{Z}_3[x]/I$ being represented as $ax^2 + bx + c + I$ by Euclidean Algorithm?

I came upon this problem in http://sites.millersville.edu/bikenaga/abstract-algebra-1/quotient-rings-of-polynomial-rings/quotient-rings-of-polynomial-rings.pdf, but I don't understand how he applied the Euclidean Algorithm to arrive at the $$ax^2 + bx + c + I$$ form for the elements in this quotient ring. Could anyone elaborate a little more? No need for the specific steps, just enough to get me started. Thanks very much!

• If $x^3+2x+1+I=I$ then $x^3\equiv-2x-1$, for example; in fact, any polynomial is equivalent to one of degree at most $2$ – J. W. Tanner Jun 3 '20 at 20:07

Divide by $$x^3 + 2x + 1$$ using the division algorithm. It's part of the statement of the theorem that the remainder has degree less than $$3$$ (i.e. less than the degree of the thing you are dividing by), hence it is of the form $$r(x) = ax^2 + bx + c$$.