If I have ring $\mathbb Z_5[x]$ ,and let $I=\langle x^2+x+2 \rangle$,then if I have to represent the elements of
$\mathbb Z_5[x]/I$ they'll be of form :
$$\{a+bx+I\mid a,b\in \mathbb Z_5\},$$
I still can't understand how can we represent them in the above form ,i.e. why did we write elements from $\mathbb Z_5[x]$ as $a+bx\mid a,b\in \mathbb Z_5$.
Please help...
We mod out by $x^2 + x + 2$, so we identify this last polynomial by $0$ in the quotient ring. This means that $x^2 + x +2 \equiv 0$ or $x^2 \equiv -x - 2 \equiv 4x + 3$, as we take coefficients in $\mathbb{Z}_5$. So also $x^3 \equiv x^2 \cdot x \equiv 4x^2 + 3x \equiv 4(-x-2) + 3x$, etc.
So all squares (and it follows from that, all higher powers of $x$ as well) can be replaced by linear terms. So a member of the quotient is represented by some polynomial (any order). but we reduce all higher than linear terms to linear terms, and get a representation as a linear polynomial.