How to find the multiplicity of a polynomial?

Question

How can you find the multiplicity of a polynomial? I have to find all $n \in \mathbb{N}$ such that $9\mid n^4+n^3-2n^2+n+4$.

The recommended method is to solve by congruence but I haven't been able to solve it.

Answer 1

Let $p(x) =x^4+x^3-3x^2+x+4$. Let $n=9k+o$ where $o$ is remainder when $n$ is divided by $9$.

Then $9|p(n)$ iff $9|p(o)$.

Proof: Since $n-o\,|\, p(n)-p(0)$ and since $9| n-o$ so $9\,|\, p(n)-p(0)$

So you have to check only for the values of $o\in\{0,1,2,...8\}$ or $o\in\{-4,-3,...3,4\}$.

$$p(0)=p(1)=4, p(2)=18, p(3)= 88, p(4) \ne 0 $$

$$p(-1) = 0, p(-2) \ne 0, p(-3)\ne 0, p(-4)\ne 0$$

so $n=9k+8$.

Answer 2

Consider the polynomial mod 9. Then $f (n)=n^4+n^3-2n^2+n+4$ is such that $f (0)=4$, $f (1)=5$, $f (2)=4$, $f (3)=7$, $f (4)=8$, $f (5)=f (-4)=7$, $f (6)=f (-3)=1$, $f (7)=f (-2)=2$, and finally $f (8)=f (-1)= 1$. Thus no integer can have $f (n) $ be a multiple of $9$, since all residues give nonzero values mod 9.