In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients.
The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer.
[1] The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see Order of a polynomial (disambiguation)).
To determine the degree of a polynomial that is not in standard form, such as
, one can put it in standard form by expanding the products (by distributivity) and combining the like terms; for example,
However, this is not needed when the polynomial is written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.
The following names are assigned to polynomials according to their degree:[2][3][4] Names for degree above three are based on Latin ordinal numbers, and end in -ic.
This should be distinguished from the names used for the number of variables, the arity, which are based on Latin distributive numbers, and end in -ary.
, is called a "binary quadratic": binary due to two variables, quadratic due to degree two.
[a] There are also names for the number of terms, which are also based on Latin distributive numbers, ending in -nomial; the common ones are monomial, binomial, and (less commonly) trinomial; thus
is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes
The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.
The equality always holds when the degrees of the polynomials are different.
Thus, the set of polynomials (with coefficients from a given field F) whose degrees are smaller than or equal to a given number n forms a vector space; for more, see Examples of vector spaces.
For polynomials over an arbitrary ring, the above rules may not be valid, because of cancellation that can occur when multiplying two nonzero constants.
over a field or integral domain is the product of their degrees:
The degree of the zero polynomial is either left undefined, or is defined to be negative (usually −1 or
It has no nonzero terms, and so, strictly speaking, it has no degree either.
[8] It is convenient, however, to define the degree of the zero polynomial to be negative infinity,
and to introduce the arithmetic rules[9] and These examples illustrate how this extension satisfies the behavior rules above: A number of formulae exist which will evaluate the degree of a polynomial function f. One based on asymptotic analysis is this is the exact counterpart of the method of estimating the slope in a log–log plot.
This formula generalizes the concept of degree to some functions that are not polynomials.
For example: The formula also gives sensible results for many combinations of such functions, e.g., the degree of
Intuitively though, it is more about exhibiting the degree d as the extra constant factor in the derivative
A more fine grained (than a simple numeric degree) description of the asymptotics of a function can be had by using big O notation.
In the analysis of algorithms, it is for example often relevant to distinguish between the growth rates of
Given a ring R, the polynomial ring R[x] is the set of all polynomials in x that have coefficients in R. In the special case that R is also a field, the polynomial ring R[x] is a principal ideal domain and, more importantly to our discussion here, a Euclidean domain.
It can be shown that the degree of a polynomial over a field satisfies all of the requirements of the norm function in the euclidean domain.
In fact, something stronger holds: For an example of why the degree function may fail over a ring that is not a field, take the following example.
This ring is not a field (and is not even an integral domain) because 2 × 2 = 4 ≡ 0 (mod 4).
Since the norm function is not defined for the zero element of the ring, we consider the degree of the polynomial f(x) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.