Editing Polynomial (section)

== Applications ==
=== Positional notation ===
{{Main|Positional notation}}
In modern positional numbers systems, such as the [[Decimal|decimal system]], the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the [[radix]] or base, in this case, {{nowrap|4 × 10<sup>1</sup> + 5 × 10<sup>0</sup>}}. As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number {{nowrap|1 × 5<sup>2</sup> + 3 × 5<sup>1</sup> + 2 × 5<sup>0</sup>}} = 42. This representation is unique.  Let ''b'' be a positive integer greater than 1.  Then every positive integer ''a'' can be expressed uniquely in the form

<math display="block">a = r_m b^m + r_{m-1} b^{m-1} + \dotsb + r_1 b + r_0,</math>
where ''m'' is a nonnegative integer and the ''r'''s are integers such that

{{math|0 < ''r''<sub>''m''</sub> < ''b''}} and {{math|0 ≤ ''r''<sub>''i''</sub> < ''b''}} for {{math|1=''i'' = 0, 1, . . . , ''m'' − 1}}.<ref>{{harvnb|McCoy|1968|p=75}}</ref>

=== Interpolation and approximation ===
{{See also|Polynomial interpolation|Orthogonal polynomials|B-spline|spline interpolation}}
The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in [[calculus]] is [[Taylor's theorem]], which roughly states that every [[differentiable function]] locally looks like a polynomial function, and the [[Stone–Weierstrass theorem]], which states that every [[continuous function]] defined on a [[compact space|compact]] [[interval (mathematics)|interval]] of the real axis can be approximated on the whole interval as closely as desired by a polynomial function.  Practical methods of approximation include [[polynomial interpolation]] and the use of [[spline (mathematics)|splines]].<ref>{{cite book |last=de Villiers |first=Johann |title=Mathematics of Approximation |publisher=Springer |year=2012 |isbn=9789491216503 |url=https://books.google.com/books?id=l5mIro_6RlUC}}</ref>

=== Other applications ===
Polynomials are frequently used to encode information about some other object. The [[characteristic polynomial]] of a matrix or linear operator contains information about the operator's [[eigenvalue]]s. The [[minimal polynomial (field theory)|minimal polynomial]] of an [[algebraic element]] records the simplest algebraic relation satisfied by that element. The [[chromatic polynomial]] of a [[graph (discrete mathematics)|graph]] counts the number of proper colourings of that graph.

The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in [[computational complexity theory]] the phrase ''[[polynomial time]]'' means that the time it takes to complete an [[algorithm]] is bounded by a polynomial function of some variable, such as the size of the input.