Editing Polynomial ring (section)

===Terminology===
Let
:<math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,</math>
be a nonzero polynomial with <math>p_m\ne 0</math>

The ''constant term'' of {{math|''p''}} is <math>p_0.</math> It is zero in the case of the zero polynomial.

The ''degree'' of {{math|''p''}}, written {{math|deg(''p'')}} is <math>m,</math> the largest {{math|''k''}} such that the coefficient of {{math|''X''{{sup|''k''}}}} is not zero.<ref>{{harvnb|Herstein|1975|p=154}}</ref>

The ''leading coefficient'' of {{math|''p''}} is <math>p_m.</math><ref>{{harvnb|Lang|2002|p=100}}</ref>

In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined,<ref>{{citation|title=Calculus Single Variable|first1=Howard|last1=Anton|first2=Irl C.|last2=Bivens|first3=Stephen|last3=Davis|publisher=Wiley |year=2012|isbn=9780470647707|page=31|url=https://books.google.com/books?id=U2uv84cpJHQC&pg=RA1-PA31}}.</ref> defined to be {{math|−1}},<ref>{{citation|title=Rational Algebraic Curves: A Computer Algebra Approach|volume=22|series=Algorithms and Computation in Mathematics|first1=J. Rafael|last1=Sendra|first2=Franz|last2=Winkler|first3=Sonia|last3=Pérez-Diaz|publisher=Springer|year=2007|isbn=9783540737247|page=250|url=https://books.google.com/books?id=puWxs7KG2D0C&pg=PA250}}.</ref> or defined to be a {{math|−∞}}.<ref>{{citation|title=Elementary Matrix Theory|publisher=Dover|first=Howard Whitley|last=Eves|author-link=Howard Eves|year=1980|isbn=9780486150277|page=183|url=https://books.google.com/books?id=ayVxeUNbZRAC&pg=PA183}}.</ref>

A ''constant polynomial'' is either the zero polynomial, or a polynomial of degree zero.

A nonzero polynomial is [[monic polynomial|monic]] if its leading coefficient is <math>1.</math>

Given two polynomials {{mvar|p}} and {{mvar|q}}, if the degree of the zero polynomial is defined to be <math>-\infty,</math> one has 
:<math>\deg(p+q) \le \max (\deg(p), \deg (q)),</math>
and, over a [[field (mathematics)|field]],  or more generally an [[integral domain]],<ref>{{harvnb|Herstein|1975|pp=155,162}}</ref>
:<math>\deg(pq) = \deg(p) + \deg(q).</math>

It follows immediately that, if {{math|''K''}} is an integral domain, then so is {{math|''K''[''X'']}}.<ref>{{harvnb|Herstein|1975|p=162}}</ref>

It follows also that, if {{math|''K''}} is an integral domain, a polynomial is a [[unit (ring theory)|unit]] (that is, it has a [[multiplicative inverse]]) if and only if it is constant and is a unit in {{mvar|K}}.

Two polynomials are [[associated element|associated]] if either one is the product of the other by a unit.

Over a field, every nonzero polynomial is associated to a unique monic polynomial.

Given two polynomials, {{mvar|p}} and {{mvar|q}}, one says that {{mvar|p}} ''divides'' {{mvar|q}}, {{mvar|p}} is a ''divisor'' of {{mvar|q}}, or {{mvar|q}} is a multiple of {{mvar|p}}, if there is a polynomial {{mvar|r}} such that {{math|1=''q'' = ''pr''}}.

A polynomial is [[irreducible polynomial|irreducible]] if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.