I need to reduce $x^3+6=7$ over $\mathbb{Z}_7$. My approach is to factor the polynomial into three irreducible (linear) factors.
This polynomial has at most three roots (up to the equivalence classes). So one can find these roots in order to then factor it into three factors of degree one.
But here's the problem: how does one solve the equation $x^3+6=7$ $\mod7$ other than by guessing?
Best Answer
$$\begin{align*} x^3+6&=7\pmod{7}\\ x^3-1&=0\pmod{7}\\ (x-1)(x^2+x+1)&=0\pmod{7}\\ (x-1)(x-2)(x-4)&=0\pmod{7} \end{align*}$$
To verify this, multiply as you normally do but remember to reduce coefficients $\pmod{7}$.