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In mathematics, the '''order of a polynomial''', relative to a particular set of [[polynomial]] [[basis function]]s spanning the [[polynomial vector space]] in which a given polynomial is included, is the highest [[Degree of a polynomial|degree]] among those basis functions used to express the polynomial.<ref name="deBoor:01">{{cite book
| last = de Boor
| first = Carl
| title = A Practical Guide to Splines
| publisher = [[Springer Publishing|Springer]]
| date = 2001
| page = 1
| isbn = 0-387-95366-3}}</ref>

== Example ==
Consider the following polynomial:

: <math>P(x) = \sum^n_{i=0} p_i x^i</math>
where <math>\left\{p_0, \ldots, p_n\right\}</math> are the polynomial coefficients and <math>\left\{1, x, \ldots, x^n\right\}</math> the set of basis functions which span the polynomial vector space.

If the polynomial coefficients are:
:<math>\left\{1, 2, 0, \ldots , 0\right\}</math>

under this polynomial vector space, <math>P(x)</math> is expressed as follows:
:<math>P(x) = 1 + 2x</math>

The degree of this polynomial would be 1. Yet, due to the set of basis functions which is used to define this polynomial, its order would be <math>n</math>.

Now, consider the same polynomial expressed in [[Lagrange polynomial|Lagrange form]].
If this polynomial is defined as a [[linear combination]] of the following set of basis functions:
:<math>\left\{ \frac{x - 0.5}{0.5-0}\cdot\frac{x-1}{0-1} ; \frac{x-0}{0.5-0}\cdot\frac{x-1}{0.5-1}; \frac{x-0}{1-0}\cdot\frac{x-0.5}{1-0.5} \right\}</math>
then, the polynomial coefficients would be:
:<math>p=\left\{1, 1.5, 2\right\}</math>.

The degree of this polynomial would still be 1, but as the highest degree of the Lagrangian basis functions is <math>2</math>, then the order of this polynomial is 2.

== See also ==
* [[Degree of a polynomial]]

== References ==
<references/>

[[Category:Polynomials]]

{{math-stub}}

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