Factoring a polynomial in a field into irreducible

Question

Factor $x^3 + 2x + 3$ into irreducible polynomials in $\mathbb{Z} _5 [x]$

This polynomial has 2 zeros mod 5: x = 2 and x = 4. But these only give me a 2 degree polynomial $x^2 - 4$ and I don't know how to find the last one.

Answer 1

This polynomial has a double root. Which one? Besides trial and error, you could also use long division to divide your polynomial by $(x-2)(x-4)$.

$x^2-4$ is not relevant here; its roots are $2$ and $3$.

Answer 2

Also you could note that the derivative is $3 x^2 + 2 = 3 (x^2 -1) = 3 (x-1)(x+1) = 3 (x - 1)(x - 4)$, so $4$ is a double root.

Answer 3

Hint $\ $ Let $\,r\,$ be the $3$rd root. By Vieta's Formulas the sum of the roots $= 0\ (=\,$ $-$coeff of $\,x^2),\,$ therefore $\, r + 2 + 4\equiv 0\pmod 5,\,$ so $\ r\equiv\, \ldots$