# Polynomial factorization in Zp

#### missavvy

1. Homework Statement
Find all p, prime for which x+2 is a factor of f(x) = 5x4 - 2x3 + 3x2 + 4x - 1 in Zp

2. Homework Equations

3. The Attempt at a Solution

So in Zp, x = p-2
I tried the first 4 primes and got the following results:
Z3, x=1, f(x) = 9 = 0
Z5, x=3, f(x) = 390 - 1 =/ 0
Z7, x=5, f(x) = 2970 - 1 =/ 0
Z11, x=9, f(x) = 31626 - 1 =/ 0

Then the only way x+2 is a factor of f(x) is if p | f(x)
However, f(x) = k(p-2) - 1 for all prime numbers, and for all k natural numbers

Then since f(x) is some multiple of p-2, -1, p never divides f(x) for all primes >= 5.

Do I have the right idea? How can I formulate this so it is more formal since it's quite sloppy..

thanks!! :)

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