Modular arithmetic - Polynomials
Sorry for bothering again, I have thee questions. I did the first two ones and just want a verification of my answers and guidance for solving the third question.
1) Find all the solutions of x^2+2x+4=0 in Z/6Z
Solution: Let f(x)=x^2+2x+4=0 then only f(2)=12 is congruent to 0 (mod 6) so x=2 is the only solution.
2) Find all prime numbers p such that x+2 is a factor of x^4+x^3+x^2-x+1 in Fp[x]
Solution: Let f(x)=x^4+x^3+x^2-x+1. If x+2 is a factor of f(x) then f(2) is congruent to 0 (mod p), p a prime number
f(2)=27 so we must find the prime numbers that divide 27, so p=3 is the only solution
3) Find all the positive integers n such that x^2+3 divides x^5-10x+12 in (Z/nZ) [x]
If someone can check my answers for question 1 and 2 and give me an answer for question 3 I would appreciate it. Thanks in advance!