How do I find the solutions of the congruence $3x^3-2x^2+x \equiv0\mod{30}$?

Question

How do I find the solutions of the congruence $3x^3-2x^2+x \equiv0\mod{30}$?

If someone could walk me through a solution so I can then attempt and do all of my other examples that would be fantastic.

Answer 1

As $30=2\cdot3\cdot5,$

$$3x^3-2x^2+x\equiv x^3-x=x(x-1)(x+1)\equiv0\pmod2$$

$$3x^3-2x^2+x\equiv x^2+x\pmod3\implies x\equiv0,-1$$

$$3x^3-2x^2+x=x(3x^2-2x+1)\equiv0\pmod5\implies$$

either $x\equiv0\pmod5$

or $0\equiv 3x^2-2x+1\pmod5$

$\implies x\equiv?.?\pmod5$

Now use CRT