Definition:Polynomial Division
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Definition
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $X$ be transcendental over $F$.
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.
From the Division Theorem for Polynomial Forms over Field:
- $\exists \map Q x, \map R x \in F \sqbrk X: \map A x = \map Q x \map B x + \map R x$
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:
- $\map A x \div \map B x = \map Q x \rem \map R x$
Notation
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.
Examples
Also see
- Definition:Division over Euclidean Domain of which this is an example, with the Euclidean valuation being the degree
- Results about polynomial division can be found here.
Linguistic Note
The verb form of the word division is divide.
Thus to divide is to perform an act of division.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division