# 2nd and 3rd degree polynomial modular arithmetic questions

I'm studying for a test and have 2 problems that I just can't figure out.

a) $3x^2+5x-2 \equiv 0 \pmod{12}$

I know that I have to seperate this to mod 3 and mod 4, but can't do much after.

b) $x^3+x+3 \equiv 0 \pmod{125}$

No idea. Probably something with $\pmod{5}$.

• Let's start by looking at the equation "modulo $3$". Trying out all three possible remainders of $x$ modulo $3$ tells us that only $x\equiv 1\pmod 3$ works.
• This implies that once we move to "modulo $6$", the only possible values of $x\pmod 6$ will be $1$ and $4$. As it turns out, both of them work modulo $6$.
• Finally, let's get up to "modulo $12$". Since we had two possible remainders modulo $6$, there are four candidates modulo $12$: $1, 4, 7, 10$. Of these, only $7$ and $10$ turn out to work; both being solutions of the original equation.
Of course, one doesn't need to restrict to primes; we could just as well start by looking at the equation modulo $4$ and check the four possibilities in order to find that $1$ and $2$ work; and then step up directly to "modulo $12$".