Consider the polynomial $f(x)= x^4-x^3+14x^ 2 + 5x+16$. Also for a prime $p$, let $\mathbb F_p$ denote the field with $p$ elements. Which of the following are always true?
Considering $f$ as a polynomial with coefficients in $\mathbb F_3$, it is a product of two irreducible factors of degree 2 over $\mathbb F_3$.
Considering $f$ as a polynomial with coefficients in $\mathbb F_7$, it has an irreducible factor of degree 3 over $\mathbb F_7$.
$f$ is a product of two polynomials of degree 2 over $\mathbb Z$.
I don't know how to factorise a polynomial over $\mathbb F_p$. Is there any algorithm to do so?