find the prime factorization of $x^3-5x^2+6x+7$ in $Z/11Z$

Question

I need to find the prime factorization of $f = x^3-5x^2+6x+7$ in $Z/11Z$

I tried the following but not sure if it is correct and if there is a better and faster way to do it.

first i tried one by one each of $x \in (1,2,..10)$ and found that 9 is a root.

then divided f by $x-9$ and got $ f = (x-9)(x^2+4x+9)$

then again trying all $x \in (1,2,...,10)$ in $x^2 +4x + 9 $ and found that it does not have any prime factorization.

so my question is - is $(x-9)(x^2+4x+9)$ the prime factorization of f?

and if so , is there a simpler way to find it?

Answer 1

There are some tricks (like the quadratic residue mentioned in the comment), but your method is the most straightforward when dealing with small degree polynomials (degree $\leq 3$) in a finite field. Keep in mind that a degree four or higher polynomial might factor into irreducible non-linear polynomials, and not have any roots.