Polynomial factorization over mod 2 and 3?

Question

i don't know what factorization over mod 2 is exactly i know what mod out of an integer is , but out of a polynomial....

so i thought that if i see some examples i can figure it out

3x^4+2x^3+5x^2+2x+2 factor mod 2

3x^4+2x^3+5x^2+2x+2 factor mod 3

3x^4+2x^3+5x^2+2x+2 factor mod 6 (i'm hoping that factored mod 6 it will remain unchanged)

Answer 1

In mod 2 x^2 = x

3x^4 + 2x^3 + 5x^2 + 2x + 2 = 3x + 2x + 5x + 2x + 2 =

= x + 0 + x + 0 + 0 = x + x = 0

In mod 3: x^(2n + m) = x^m (x^2 = 1 for a non-zero x)

3x^4 + 2x^3 + 5x^2 + 2x + 2 = 0 + 2x^3 + 2x^2 + 2x + 2 =

= 2x + 2x^2 + 2x + 2 = 2x^2 + 4x + 2 = 2(x^2 + 2x + 1) =

= 2(x + 1)^2 = (2x + 1)(x + 1)

For mod 6 change nothing, for there's no power greater than 5=6-1, and no coefficient is greater than 6.

Answer 2

3x^4+2x^3+5x^2+2x+2

i don't no what u mean by polynomial