Show that polynomial is reducible

Question

Show that $p(x)$ = $x^3 + 3x^2 + 2x + 4$ is reducible in $\mathbb{Z}$$/$$7$$\mathbb{Z}$

Is the approach for this to factor it and then find a root? I'm a little confused on how to start. Any tips appreciated.

Answer 1

Since $p$ has degree $3$, it is reducible iff it has a root. Since $\mathbb Z / 7 \mathbb Z$ is a finite field, we can test every element to see if it's a root or not.

Answer 2

Another way:

\begin{align}p(x) &= x^3 + 3x^2 + 2x + 4\\ & = x^3 - 4x^2 + 2x + 4\\ & = x^3 - 2x^2 - 2x^2 + 4x - 2x + 4\\ & = x^2(x - 2) - 2x(x - 2) - 2(x - 2)\\ & = (x^2 - 2x - 2)(x - 2). \end{align}