what is 3x^2 + 2x + 4 when factored in mod 5? What are the steps?

Question

Is there a specific way to solve it, if so how? and how do many people get multiple different answers. Please help; very confused

Answer 1

A quadratic (or cubic) polynomial factors over a field (such as the integers mod 5) if and only if it has a root; that is, if $a$ is a root of your polynomial, then $(x - a)$ must be one of its factors. Since there are only five possibilities for roots when you're working mod 5, it's easy to determine whether there are roots--just check all five of them.

Answer 2

My first impulse was to multiply the polynomial by $2$ (the inverse of $3\pmod5$) to make it monic:

$2(3x^2+2x+4)=6x^2+4x+8\equiv \color{blue}{x^2+4x+3}\pmod5.$

Now can you factor it?

Multiply the factors again, this time by $3,$

to get back to the original polynomial $3x^2+2x+4\pmod5$.

People may get answers that look different,

because $-1$ and $4$ are equivalent $\pmod 5$, for example.