Dividing a polynomial in a field $F_5$

Question

I'm dividing $2x^3 + x^2 - 3x + 1$ in $F_5$ with x-3, with long division. I'm not sure if I should be taking modulus at every step?

For example, first I multiply (x-3) with $2x^2$, yield $2x^3-6x^2$. Subtracting this with $2x^3 + x^2 - 3x + 1$, I get $7x^2 - 3x +1$. Should I mod the 7 with 5, or mod the remainder?

EDIT:

Taking the modulus at each step yielded $2x^2+2x+3$, with a remainder of 10 (which is basically 0). Is this correct?

Answer 1

Yes, that is correct. Taking modulus at every step vs in the end, both work as well in this case, as 5 is a prime. Otherwise, division is not well behaved in general in modular arithmetic.