# Order of a polynomial

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In mathematics, the **order of a polynomial**, relative to a particular set of polynomial basis functions spanning the polynomial vector space in which a given polynomial is included, is the highest degree among those basis functions used to express the polynomial.^{[1]}

## 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 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

## References

- ↑ de Boor, Carl (2001).
*A Practical Guide to Splines*. Springer. p. 1. ISBN 0-387-95366-3.