abstract algebra - Write $x^3 + 2x+1$ as a product of linear polynomials over some extension field of $mathbb{Z}_3$

Write $x^3 + 2x+1$ as a product of linear polynomials over some extension field of $\mathbb{Z}_3$

Long division seems to be taking me nowhere, If $\beta$ is a root in some extension then using long division one can write

$$x^3 + 2x+1 = (x-\beta) (x^2+ \beta x+ \beta^2 + 2)$$

Here is a similar question

Suppose that $\beta$ is a zero of $f(x)=x^4+x+1$ in some field extensions of $E$ of $Z_2$.Write $f(x)$ as a product of linear factors in $E[x]$

Is there a general method to approach such problems or are they done through trail and error method.

I haven't covered Galois theory and the problem is from field extensions chapter of gallian, so please avoid Galois theory if possible.

Answer

If you want to continue the way you started, i.e. with

$$x^3 + 2x+1 = (x-\beta) (x^2+ \beta x+ \beta^2 + 2)$$
you can try to find the roots of the second factor by using the usual method for quadratics, adjusted for characteristic 3. I'll start it so you know what I mean:

To solve $x^2+ \beta x+ \beta^2 + 2=0$, we can complete the square, noting that $2\beta + 2\beta = \beta$ and $4\beta^2=\beta^2$in any extension of $\mathbb{Z}_3$ (since $4\equiv 1$).

$$x^2+ \beta x+ \beta^2 + 2=(x+2\beta)^2 + 2=0$$
But this is easy now since this is the same as
$$(x+2\beta)^2 = 1$$

which should allow you to get the remaining roots.