Find all solutions to $x^3-x^2+2x-2=0 \pmod {11}$

Question

I get that $11$ is a small number and that I could maybe do this by inspection, but I was wondering if there was a more intelligent approach?

Answer 1

$$x^3-x^2+2x-2\equiv0 \pmod {11}$$

$$(x^2+2)(x-1)\equiv0\pmod{11}$$

Since $11$ is prime,

$$x-1\equiv 0\pmod{11}\implies x\equiv 1 \pmod{11}$$

OR

$$x^2\equiv-2\pmod{11}$$

Can you finish the last conrguence by yourself?

Answer 2

Hint: you could try to factor $$\begin{align} x^2(x-1) + 2(x-1) &= 0 \pmod{11}\\ (x^2 +2)(x-1) &= 0 \pmod{11} \end{align} $$