Editing Polynomial (section)

== Polynomial functions ==
<!-- "Polynomial function" redirects here -->
{{See also|Ring of polynomial functions}}

A ''polynomial function'' is a function that can be defined by [[#evaluation|evaluating]] a polynomial. More precisely, a function {{math|''f''}} of one [[argument of a function|argument]] from a given domain is a polynomial function if there exists a polynomial
<math display="block">a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 </math>
that evaluates to <math>f(x)</math> for all {{mvar|x}} in the [[domain of a function|domain]] of {{mvar|f}} (here, {{math|''n''}} is a non-negative integer and {{math|''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>''}} are constant coefficients).{{sfn|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/38/mode/1up?view=theater&q=polynomial 38]}}
Generally, unless otherwise specified, polynomial functions have [[complex number|complex]] coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also [[restriction of a function|restricted]] to the reals, the resulting function is a [[real function]] that maps reals to reals.

For example, the function {{math|''f''}}, defined by
<math display="block"> f(x) = x^3 - x,</math>
is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in
<math display="block">f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.</math>
According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression <math>\left(\sqrt{1-x^2}\right)^2,</math> which takes the same values as the polynomial <math>1-x^2</math> on the interval <math>[-1,1]</math>, and thus both expressions define the same polynomial function on this interval.

Every polynomial function is [[continuous function|continuous]], [[smooth function|smooth]], and [[entire function|entire]].

{{anchor|evaluation}}The [[polynomial evaluation|'''evaluation''']] of a polynomial is the computation of the corresponding polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions.

For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using [[Horner's method]], which consists of rewriting the polynomial as
<math display="block">(((((a_n x + a_{n-1})x + a_{n-2})x + \dotsb + a_3)x + a_2)x + a_1)x + a_0.</math>

=== Graphs ===
<div class="floatright">
<gallery perrow="2" widths="120px" heights="120px">
File:Algebra1 fnz fig037 pc.svg|Polynomial of degree 0:<br/><small>{{math|''f''(''x'') {{=}} 2}}</small>
File:Fonction de Sophie Germain.png|Polynomial of degree 1:<br/><small>{{math|''f''(''x'') {{=}} 2''x'' + 1}}</small>
File:Polynomialdeg2.svg|Polynomial of degree 2:<br/><small>{{math|''f''(''x'') {{=}} ''x''<sup>2</sup> − ''x'' − 2}}<br/>{{math|{{=}} (''x'' + 1)(''x'' − 2)}}</small>
File:Polynomialdeg3.svg|Polynomial of degree 3:<br/><small>{{math|''f''(''x'') {{=}} ''x''<sup>3</sup>/4 + 3''x''<sup>2</sup>/4 − 3''x''/2 − 2}}<br/>{{math|{{=}} 1/4 (''x'' + 4)(''x'' + 1)(''x'' − 2)}}</small>
File:Polynomialdeg4.svg|Polynomial of degree 4:<br/><small>{{math|''f''(''x'') {{=}} 1/14 (''x'' + 4)(''x'' + 1)(''x'' − 1)(''x'' − 3) <br/>+ 0.5}}</small>
File:Quintic polynomial.svg|Polynomial of degree 5:<br/><small>{{math|''f''(''x'') {{=}} 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1)(''x'' − 1)<br/>(''x'' − 3) + 2}}</small>
File:Sextic Graph.svg|Polynomial of degree 6:<br/><small>{{math|''f''(''x'') {{=}} 1/100 (''x''<sup>6</sup> − 2''x'' <sup>5</sup> − 26''x''<sup>4</sup> + 28''x''<sup>3</sup>}}<br/>{{math|+ 145''x''<sup>2</sup> − 26''x'' − 80)}}</small>
File:Septic graph.svg|Polynomial of degree 7:<br/><small>{{math|''f''(''x'') {{=}} (''x'' − 3)(''x'' − 2)(''x'' − 1)(''x'')(''x'' + 1)(''x'' + 2)}}<br/>{{math|(''x'' + 3)}}</small>
</gallery>
</div>
A polynomial function in one real variable can be represented by a [[graph of a function|graph]].
<ul>
<li>
The graph of the zero polynomial
{{block indent|{{math|1=''f''(''x'') = 0}}}} is the {{math|''x''}}-axis.
</li>
<li>
The graph of a degree 0 polynomial
{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub>}}, where {{math|''a''<sub>0</sub> ≠ 0}},}} is a horizontal line with {{nowrap|{{math|''y''}}-intercept {{math|''a''<sub>0</sub>}}}}
</li>
<li>
The graph of a degree 1 polynomial (or linear function)
{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x''}}, where {{math|''a''<sub>1</sub> ≠ 0}},}} is an oblique line with {{nowrap|{{math|''y''}}-intercept {{math|''a''<sub>0</sub>}}}} and [[slope]] {{math|''a''<sub>1</sub>}}.
</li>
<li>
The graph of a degree 2 polynomial
{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup>}}, where {{math|''a''<sub>2</sub> ≠ 0}}}} is a [[parabola]].
</li>
<li>
The graph of a degree 3 polynomial
{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ''a''<sub>3</sub>''x''<sup>3</sup>}}, where {{math|''a''<sub>3</sub> ≠ 0}}}} is a [[cubic equation|cubic curve]].
</li>
<li>
The graph of any polynomial with degree 2 or greater
{{block indent|{{math|1=''f''(''x'') = ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ⋯ + ''a''<sub>''n''</sub>''x''<sup>''n''</sup>}}, where {{math|''a''<sub>''n''</sub> ≠ 0 and ''n'' ≥ 2}}}} is a continuous non-linear curve.
</li>
</ul>

A non-constant polynomial function [[infinity#Calculus|tends to infinity]] when the variable increases indefinitely (in [[absolute value]]). If the degree is higher than one, the graph does not have any [[asymptote]]. It has two [[parabolic branch]]es with vertical direction (one branch for positive ''x'' and one for negative ''x'').

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.