Division algorithm for polynomials

Statement

Let $R$ be a commutative ring, let $f \in R[x]$ and let $g \in R[x]$ be a monic polynomial. Then there are unique polynomials $q, r \in R[x]$ such that $f = q g + r$ and $\deg r < \deg g$.

Remarks

The polynomials $q$ and $r$ are the quotient and the remainder. Long division of polynomials computes them. Over a field $K$, every nonzero $g$ is a unit times a monic polynomial. So the statement holds for every nonzero $g \in K[x]$. For example, $x^3 + 1 = (x^2 - x + 1)(x + 1) + 0$ and $x^2 + 1 = (x - 1)(x + 1) + 2$. The statement fails for a divisor that is not monic: in $\mathbb{Z}[x]$, no $q$ and $r$ with $\deg r < 1$ give $x = q \cdot 2x + r$.

Explanations

L1 human

This is the polynomial version of division with remainder for integers. You can divide by a monic polynomial and get a remainder of smaller degree. The degree plays the role that the size of a number plays for integers.