How to solve polynomial Congruence

Question

$21x^5 + 10x^3 + 14x \equiv 0 \pmod{3}$.

I want to use flt and crt to solve this but I am not sure where to start. Do you have any tips?

Answer 1

The given equation is equivalent mod $3$ to (using that $\Bbb Z_3$ is a field)

$$x^3+2x=0\iff x(x^2+2)=0\iff x=0\lor x=\pm 1$$

Answer 2

$$21x^5+10x^3+14x\equiv x^3-x\pmod3$$

But $x^3-x=(x-1)x(x+1)$ being product of three consecutive integer is always divisible by $3$

Hence $21x^5+10x^3+14x\equiv0\pmod3$ is an examaple of Identical Congruence