polynomial roots

**sfspitfire23** · December 4th 2009, 05:15 PM

Two questions that I know the answers to

1. How many polynomials of deg 3 are there in $\mathbb{Z}_5[X]$ ? The answer is 500.

2. Find the roots of $(x-4)(x-5)$ in $\mathbb{Z}_6$ . 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 0-4...

**Bruno J.** · December 4th 2009, 05:50 PM

For $1)$ , notice that we only have $4$ choices for the coefficient of $x^3$ because it has to be nonzero. We have $5$ choices for all the other coefficients, hence we get $4 \times 5^3 = 500$ .

For question $2)$ , just try all the elements of $\mathbb{Z}_6$ !

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