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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!

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    $\begingroup$ 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 $2$ $\endgroup$ Commented Jun 3, 2020 at 20:07

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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 $r(x) = ax^2 + bx + c$.

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