
polynomial roots
Two questions that I know the answers to
1. How many polynomials of deg 3 are there in $\displaystyle \mathbb{Z}_5[X]$? The answer is 500.
2. Find the roots of $\displaystyle (x4)(x5)$ in $\displaystyle \mathbb{Z}_6$. Answer= 4,5,1,2
I know for question 2 the 1,2 comes from the fact that its in Z5, but I cant seem to apply that fact to get 1,2 myself. For question 1, what is the methodology of finding 500? I know that each coefficient can only be 04...

For $\displaystyle 1)$, notice that we only have $\displaystyle 4$ choices for the coefficient of $\displaystyle x^3$ because it has to be nonzero. We have $\displaystyle 5$ choices for all the other coefficients, hence we get $\displaystyle 4 \times 5^3 = 500$.
For question $\displaystyle 2)$, just try all the elements of $\displaystyle \mathbb{Z}_6$!