Editing Polynomial (section)

== Classification ==
{{Further|Degree of a polynomial}}
The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any term with nonzero coefficient.<ref name=":2">{{Cite web|title=Polynomials {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/polynomials/|access-date=2020-08-28|website=brilliant.org|language=en-us}}</ref> Because {{math|''x'' {{=}} ''x''<sup>1</sup>}}, the degree of an indeterminate without a written exponent is one.

{{anchor|constant polynomial}}
A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a [[constant term]] and a '''constant polynomial'''.{{efn|This terminology dates from the time when the distinction was not clear between a polynomial and the function that it defines: a constant term and a constant polynomial define [[constant function]]s.{{citation needed|date=July 2020}}}} The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below).<ref name=Barbeau-2003-pp1-2>{{harvnb|Barbeau|2003|pp=[https://books.google.com/books?id=CynRMm5qTmQC&pg=PA1 1]–2}}</ref>

For example:
<math display="block"> -5x^2y </math>
is a term. The coefficient is {{math|−5}}, the indeterminates are {{math|''x''}} and {{math|''y''}}, the degree of {{math|''x''}} is two, while the degree of {{math|''y''}} is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is {{math|2 + 1 {{=}} 3}}.

Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
<math display="block">\underbrace{_\,3x^2}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}} \underbrace{-_\,5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}} \underbrace{+_\,4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}. </math>
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.

{{anchor|linear polynomial}}Polynomials of small degree have been given specific names. A polynomial of degree zero is a ''constant polynomial'', or simply a ''constant''. Polynomials of degree one, two or three are respectively ''linear polynomials,'' ''[[quadratic polynomial]]s'' and ''cubic polynomials''.<ref name=":2" /> For higher degrees, the specific names are not commonly used, although ''quartic polynomial'' (for degree four) and ''quintic polynomial'' (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term {{math|2''x''}} in {{math|''x''<sup>2</sup> + 2''x'' + 1}} is a linear term in a quadratic polynomial.

{{anchor|zero polynomial}}The polynomial 0, which may be considered to have no terms at all, is called the '''zero polynomial'''. Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).<ref>{{MathWorld |urlname=ZeroPolynomial |title=Zero Polynomial}}</ref> The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of [[root of a function|roots]]. The graph of the zero polynomial, {{math|''f''(''x'') {{=}} 0}}, is the ''x''-axis.

In the case of polynomials in more than one indeterminate, a polynomial is called ''homogeneous'' of {{nowrap|degree {{math|''n''}}}} if ''all'' of its non-zero terms have {{nowrap|degree {{math|''n''}}}}. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined.{{efn|In fact, as a [[homogeneous function]], it is homogeneous of ''every'' degree.{{citation needed|date=July 2020}}}} For example, {{math|''x''<sup>3</sup>''y''<sup>2</sup> + 7''x''<sup>2</sup>''y''<sup>3</sup> − 3''x''<sup>5</sup>}} is homogeneous of degree 5. For more details, see [[Homogeneous polynomial]].

The [[commutative law]] of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of {{math|''x''}}", with the term of largest degree first, or in "ascending powers of {{math|''x''}}". The polynomial {{math|3''x''<sup>2</sup> − 5''x'' + 4}} is written in descending powers of {{math|''x''}}. The first term has coefficient {{math|3}}, indeterminate {{math|''x''}}, and exponent {{math|2}}. In the second term, the coefficient {{nowrap|is {{math|−5}}}}. The third term is a constant. Because the ''degree'' of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.<ref>{{harvnb|Edwards|1995|p=[https://books.google.com/books?id=ylFR4h5BIDEC&pg=PA78 78]}}</ref>

Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the [[distributive law]], into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0.<ref name="Edwards-1995-p47"/> Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a [[monomial]],{{efn|Some authors use "monomial" to mean "[[monic polynomial|monic]] monomial". See {{cite book |first=Anthony W. |last=Knapp |title=Advanced Algebra: Along with a Companion Volume Basic Algebra |page=457 |year=2007 |publisher=Springer |isbn=978-0-8176-4522-9}}}} a two-term polynomial is called a [[binomial (polynomial)|binomial]], and a three-term polynomial is called a [[trinomial]].

{{anchor|real polynomial|complex polynomial|integer polynomial}}A '''real polynomial''' is a polynomial with [[real number|real]] coefficients. When it is used to define a [[function (mathematics)|function]], the [[domain (function)|domain]] is not so restricted. However, a '''real polynomial function''' is a function from the reals to the reals that is defined by a real polynomial. Similarly, an '''integer polynomial''' is a polynomial with [[integer]] coefficients, and a '''complex polynomial''' is a polynomial with [[complex number|complex]] coefficients.

{{anchor|univariate|bivariate|Number of variables|Multivariate polynomial}}A polynomial in one indeterminate is called a '''univariate polynomial''', a polynomial in more than one indeterminate is called a '''multivariate polynomial'''. A polynomial with two indeterminates is called a '''bivariate polynomial'''.<ref name=":1" /> These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials (which may result from the subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as ''bivariate'', ''trivariate'', and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is also common to say simply "polynomials in {{math|''x'', ''y''}}, and {{math|''z''}}", listing the indeterminates allowed.