Definition:Divisor of Polynomial

Definition

Let $D$ be an integral domain.

Let $D \sqbrk x$ be the polynomial ring in one variable over $D$.

Let $f, g \in D \sqbrk x$ be polynomials.

Then:

$f$ divides $g$

$f$ is a divisor of $g$

$g$ is divisible by $f$

if and only if:

$\exists h \in D \sqbrk x : g = f h$

This is denoted:

$f \divides g$

Notation

The conventional notation for $x$ is a divisor of $y$ is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by Knuth etc.

From Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.):

The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.

An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.

Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \ $ or similar to denote non-divisibility.

Examples

Arbitrary Example $1$

The expressions:

$x - 1$

$x + 2$

are divisors of the polynomial $x^2 + x - 2$

Arbitrary Example $2$

The polynomial:

$2 x^2 + 1$

can be expressed in terms of its divisors as:

$2 \paren {x^2 + 1}$

Arbitrary Example $3$

The polynomial:

$x^2 - 2$

can be expressed in terms of its divisors as:

$\paren {x + \sqrt 2} \paren {x - \sqrt 2}$

Arbitrary Example $4$

The polynomial:

$x^2 + y^2$

can be expressed in terms of its divisors as:

$\paren {x + i y} \paren {x - i y}$

Also known as

A divisor can also be referred to as a factor.

Also see

• Definition:Divisor of Ring Element: see Ring of Polynomial Forms is Integral Domain

• Results about divisors of polynomials can be found here.

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 37$. Roots of Polynomials: Theorem $70$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisible
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisor
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisible
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisor: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor: 1.