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In mathematics, the order of a polynomial may have several meanings, depending on the context.

Order has been used to denoted the degree of a polynomial. Nowadays, this terminology is rarely used.
Order may refer to the order of the polynomial, viewed as a power series, that is the degree of its nonzero monomial of lowest degree.

The meaning in this article.

In this article, 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:

\( P(x) = \sum^n_{i=0} p_i x^i \)

where \( \left(p_0, \ldots, p_n\right) \) are the polynomial coefficients and \( \left\{1, x, \ldots, x^n\right\} \) the set of basis functions which span the polynomial vector space.

If the polynomial coefficients are:

\( \left(1, 2, 0, \ldots, 0\right) \)

under this polynomial vector space, P(x) is expressed as follows:

P(x) = 1 + 2x

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 n.

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:

\( \left\{ \frac{x - 0.5}{0.5-0}\cdot\frac{x-1}{0-1} ; \frac{x-0}{0.5-0}\cdot\frac{x-1}{0.5-1}; \frac{x-0}{1-0}\cdot\frac{x-0.5}{1-0.5} \right\} \)

then, the polynomial coefficients would be:

\( p=\left\{1, 1.5, 2\right\}. \)

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

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Order of a polynomial