Definition:Polynomial Division

Definition

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Let $\map A x$ and $\map B x$ be polynomials in $F \sqbrk X$ such that the degree of $B$ is non-zero.

$\exists \map Q x, \map R x \in F \sqbrk X: \map A x = \map Q x \map B x + \map R x$

such that:

$0 \le \map \deg R < \map \deg B$

where $\deg$ denotes the degree of a polynomial.

The process of finding $\map Q x$ and $\map R x$ is known as polynomial division of $\map A x$ by $\map B x$, and we write:

$\map A x \div \map B x = \map Q x \rem \map R x$

The operation of division can be denoted as:

$a / b$, which is probably the most common in the general informal context

$\dfrac a b$, which is the preferred style on $\mathsf{Pr} \infty \mathsf{fWiki}$

$a : b$, which is usually used when discussing ratios

$a \div b$, which is rarely seen outside grade school, but can be useful in contexts where it is important to be specific.

The verb form of the word division is divide.

Thus to divide is to perform an act of division.