# Can the concept of congruence be applied to the remainder of a polynomial division?

I know this is a very simple question, so please I apologize but I am not familiar with it:

Can the concept of (modular arithmetic) congruence be applied to the remainder of a polynomial division? (e.g. when the divisor is a polynomial of at least degree 1?

For instance, $\frac{x^2-1}{x+1}=x-1$ with remainder = $0$. Then:

Does it means that $x^2-1 \equiv 0 \pmod {x+1}?$

If so, what happens if the remainder is not an integer? If I am not wrong the remainder of the polynomial division can be for instance a fraction as well so I am not sure what happens (what is the meaning) when the remainder is not exactly an integer.

I have seen that there is a tag in MSE about polynomial congruences and did some searching but the questions are very few and different, and some of them not related with a question regarding a division in which the divisor is also a polynomial.

If somebody could provide me a basic explanation as a starting point (or an online resource to learn a minimum base) would be very appreciated.

Thank you!

(If the question can be rewritten to express it better please feel free to modify it, if it is duplicated I will remove it, just let me know)

You were probably taught that two integers $a$ and $b$ are congruent modulo another integer $n$ if $n$ divides $a-b$. Well, the same applies to many other objects, such as polynomials but also matrices for example, which can be added and multiplied (the mathematical term is, "in any ring").
Two polynomials $P$ and $Q$ (with, say, integer or real or complex coefficients) are congruent modulo another polynomial $N$ if $N$ divides $P-Q$. In the special case where $Q = 0$, this means that $N$ divides $P$, which is what you have here.
• easy to follow, very appreciated! Please may I ask you? then how would be the correct way of expressing that in my sample? e.g. would it be "$x^2-1$ is congruent 0 modulo $(x-1)$" or "$x^2-1$ and $0$ are congruent modulo $x-1$"? – iadvd Jun 1 '15 at 9:29