I'm currently trying to understand polynomial division in abstract algebra. How is it possible for a polynomial with negative coefficients, say $f(x)=x^{4}-3x^{3}+2x^{3}+4x-1$ be a member of the set of polynomials in $\mathbb{Z}_{5}[x]$? This comes as a problem to me since as far as I know, negative numbers are not members of $\mathbb{Z}_{5}$, but if some arithmetic will result in negative numbers, it should undergo modular arithmetic. For example, note that 2 and 3 are elements of $\mathbb{Z}_{5}$. So for subtraction,
$2-3=-1=4\pmod 5=4\in\mathbb{Z}_{5}$.
But in Fraleigh's example for polynomial division, he left it as it is, i.e. $2x^{2}-3x^{2}=-x^{2}$. Shouldn't it be $4x^{2}$ since we're working on $\mathbb{Z}_{5}$?