## Number of polynomials

Determine how many polynomials $$p$$ in $$\mathbb Z_5$$ of degree three satisfy $$p(3)=2$$. Describe all such polynomials.
For a possible $$p(x)=a_3x^3+a_2x^2+a_1x+a_0$$ we have four choices for $$a_3$$, namely: $$1,…,4$$; for $$a_2$$ and $$a_1$$ five choices: $$0,…,4$$; and the value of $$a_0$$ can be determined when $$x$$ is subtituted by 3. In total there are hundred of such polynomials.
$$\{a_3x^3+a_2x^2+a_1x+(3a_3+a_2+2a_1+2): a_3\in\{1,…,4\}, a_2, a_1\in\{0,…,4\}\}$$.