Does $x^3-x+3$ have any roots in $\mathbb{F}_5$?

Question

Does $x^3-x+3$ have any roots in $\mathbb{F}_5$? I don't think it does but the only way I know to check is by trial and error and there's only so many factors I can try before my hand cramps up.

Answer 1

Note $$x^3-x=x\left(x^2-1\right) = (x-1) x (x+1)$$ hence $-1, 0, 1$ will not work as you need $x^3-x=-3 \equiv 2 \pmod 5$, check the other values $\pm 2$?