In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the degree of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
is the highest of the degrees of the polynomial's
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s (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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. For a
univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.
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)).
For example, the polynomial
which can also be written as
has three terms. The first term has a degree of 5 (the sum of the
powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term.
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,
is of degree 1, even though each summand has degree 2. 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.
Names of polynomials by degree
The following names are assigned to polynomials according to their degree:
* Special case –
zero (see
Degree of the zero polynomial below)
* Degree 0 – non-zero
constant
* Degree 1 –
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
* Degree 2 –
quadratic
* Degree 3 –
cubic
* Degree 4 –
quartic (or, if all terms have even degree,
biquadratic)
* Degree 5 –
quintic
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
* Degree 6 –
sextic
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.
A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More preci ...
(or, less commonly, hexic)
* Degree 7 –
septic (or, less commonly, heptic)
* Degree 8 – octic
* Degree 9 – nonic
* Degree 10 – decic
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''. For example, a degree two polynomial in two variables, such as
, is called a "binary quadratic": ''binary'' due to two variables, ''quadratic'' due to degree two. 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
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
'', ''
binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
*Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
* ...
'', and (less commonly) ''
trinomial''; thus
is a "binary quadratic binomial".
Examples
The polynomial
is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes
, with highest exponent 3.
The polynomial
is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving
, with highest exponent 5.
Behavior under polynomial operations
The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.
Addition
The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; that is,
:
and
.
For example, the degree of
is 2, and 2 ≤ max.
The equality always holds when the degrees of the polynomials are different. For example, the degree of
is 3, and 3 = max.
Multiplication
The degree of the product of a polynomial by a non-zero
scalar is equal to the degree of the polynomial; that is,
:
.
For example, the degree of
is 2, which is equal to the degree of
.
Thus, the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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
This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis.
''Notation''. Let ''F'' denote an arbitrary field such as the real numbers R or the complex numbers C ...
.
More generally, the degree of the product of two polynomials over a
field or an
integral domain is the sum of their degrees:
:
.
For example, the degree of
is 5 = 3 + 2.
For polynomials over an arbitrary
ring, the above rules may not be valid, because of cancellation that can occur when multiplying two nonzero constants. For example, in the ring
of
integers modulo 4, one has that
, but
, which is not equal to the sum of the degrees of the factors.
Composition
The degree of the composition of two non-constant polynomials
and
over a field or integral domain is the product of their degrees:
:
.
For example:
*If
,
, then
, which has degree 6.
Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in
,
, but
.
Degree of the zero polynomial
The degree of the
zero polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
is either left undefined, or is defined to be negative (usually −1 or
).
Like any constant value, the value 0 can be considered as a (constant) polynomial, called the
zero polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is usually undefined. The propositions for the degree of sums and products of polynomials in the above section do not apply, if any of the polynomials involved is the zero polynomial.
It is convenient, however, to define the degree of the zero polynomial to be ''negative infinity'',
and to introduce the arithmetic rules
[Axler (1997) gives these rules and says: "The 0 polynomial is declared to have degree so that exceptions are not needed for various reasonable results." (p. 64)]
:
and
:
These examples illustrate how this extension satisfies the
behavior rules above:
*The degree of the sum
is 3. This satisfies the expected behavior, which is that
.
*The degree of the difference
is
. This satisfies the expected behavior, which is that
.
*The degree of the product
is
. This satisfies the expected behavior, which is that
.
Computed from the function values
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
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y=ax^k – appear ...
.
This formula generalizes the concept of degree to some functions that are not polynomials.
For example:
*The degree of the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
,
, is −1.
*The degree of the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
,
, is 1/2.
*The degree of the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
,
, is 0.
*The degree of the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
,
, is
The formula also gives sensible results for many combinations of such functions, e.g., the degree of
is
.
Another formula to compute the degree of ''f'' from its values is
:
;
this second formula follows from applying
L'Hôpital's rule to the first formula. Intuitively though, it is more about exhibiting the degree ''d'' as the extra constant factor in the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of
.
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
and
, which would both come out as having the ''same'' degree according to the above formulae.
Extension to polynomials with two or more variables
For polynomials in two or more variables, the degree of a term is the ''sum'' of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial ''x''
2''y''
2 + 3''x''
3 + 4''y'' has degree 4, the same degree as the term ''x''
2''y''
2.
However, a polynomial in variables ''x'' and ''y'', is a polynomial in ''x'' with coefficients which are polynomials in ''y'', and also a polynomial in ''y'' with coefficients which are polynomials in ''x''. The polynomial
:
has degree 3 in ''x'' and degree 2 in ''y''.
Degree function in abstract algebra
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
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers ...
.
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. That is, given two polynomials ''f''(''x'') and ''g''(''x''), the degree of the product ''f''(''x'')''g''(''x'') must be larger than both the degrees of ''f'' and ''g'' individually. 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. Let ''R'' =
, the ring of integers
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
4. This ring is not a field (and is not even an
integral domain) because 2 × 2 = 4 ≡ 0 (mod 4). Therefore, let ''f''(''x'') = ''g''(''x'') = 2''x'' + 1. Then, ''f''(''x'')''g''(''x'') = 4''x''
2 + 4''x'' + 1 = 1. Thus deg(''f''⋅''g'') = 0 which is not greater than the degrees of ''f'' and ''g'' (which each had degree 1).
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.
See also
*
Abel–Ruffini theorem
*
Fundamental theorem of algebra
Notes
References
*
*
*
*
*
*
*
External links
Polynomial Order Wolfram MathWorld
{{DEFAULTSORT:Degree of a Polynomial
Polynomials