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The **degree of a polynomial ** is the highest of the degrees of its 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. 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 expressed 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.

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, *x*, is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

In mathematics, a **monomial** is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

In elementary mathematics, a **variable** is a symbol, commonly a single letter, that represents a number, called the *value* of the variable, which is either arbitrary, not fully specified, or unknown. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation for the variables that represent them.

- Names of polynomials by degree
- Other examples
- Behavior under polynomial operations
- Addition
- Scalar multiplication
- Multiplication
- Composition
- Degree of the zero polynomial
- Computed from the function values
- Extension to polynomials with two or more variables
- Degree function in abstract algebra
- Notes
- References
- External links

To determine the degree of a polynomial that is not in standard form (for example:), one has to put it first 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 expressed as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.

Look up in Wiktionary, the free dictionary. Appendix:English polynomial degrees |

The following names are assigned to polynomials according to their degree:^{ [1] }^{ [2] }^{ [3] }

- Special case – zero (see § Degree of the zero polynomial below)
- Degree 0 – non-zero constant
^{ [4] } - Degree 1 – linear
- Degree 2 – quadratic
- Degree 3 – cubic
- Degree 4 – quartic (or, if all terms have even degree, biquadratic)
- Degree 5 – quintic
- Degree 6 – sextic (or, less commonly, hexic)
- Degree 7 – septic (or, less commonly, heptic)

In mathematics, a **constant function** is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value .

In mathematics, the term **linear function** refers to two distinct but related notions:

In algebra, a **cubic function** is a function of the form

For higher degrees, names have sometimes been proposed,^{ [5] } but they are rarely used:

- Degree 8 – octic
- Degree 9 – nonic
- Degree 10 – decic

In algebra, an **octic equation** is an equation of the form

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.^{ [lower-alpha 1] } 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 "binary quadratic binomial".

- The polynomial is a nonic polynomial
- The polynomial is a cubic polynomial
- The polynomial is a quintic polynomial (as the are cancelled out)

The canonical forms of the three examples above are:

- for , after reordering, ;
- for , after multiplying out and collecting terms of the same degree, ;
- for , in which the two terms of degree 8 cancel, .

The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.^{ [6] }

The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; the equality always holds when the degrees of the polynomials are different i.e.

- .
- .

E.g.

- The degree of is 3. Note that 3 ≤ max{3, 2}
- The degree of is 2. Note that 2 ≤ max{3, 3}

The degree of the product of a polynomial by a non-zero scalar is equal to the degree of the polynomial, i.e.

- .

E.g.

- The degree of is 2, just as the degree of .

Note that for polynomials over a ring containing divisors of zero, this is not necessarily true. For example, in , , but .

The set of polynomials with coefficients from a given field F and degree smaller than or equal to a given number n thus forms a vector space. (Note, however, that this set is not a ring, as it is not closed under multiplication, as is seen below.)

The degree of the product of two polynomials over a field or an integral domain is the sum of their degrees:

- .

E.g.

- The degree of is 3 + 2 = 5.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in , , but .

The degree of the composition of two non-constant polynomials and over a field or integral domain is the product of their degrees:

- .

E.g.

- If , , then , which has degree 6.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in , , but .

The degree of the zero polynomial is either left undefined, or is defined to be negative (usually −1 or ).^{ [7] }

Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is 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.^{ [8] }

It is convenient, however, to define the degree of the zero polynomial to be *negative infinity*, and introduce the arithmetic rules^{ [9] }

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 .

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 degree of the multiplicative inverse, , is −1.
- The degree of the square root, , is 1/2.
- The degree of the logarithm, , is 0.
- The degree of the exponential function, , is

Note that 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 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.

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*.

Given a ring R, the polynomial ring R[*x*] is the set of all polynomials in *x* that have coefficients chosen from R. In the special case that R is also a field, then 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. 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:

- deg(
*f*(*x*)*g*(*x*)) = deg(*f*(*x*)) + deg(*g*(*x*))

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 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.

- ↑ For simplicity, this is a homogeneous polynomial, with equal degree in both variables separately.

- ↑ "Names of Polynomials". 25 November 1997. Retrieved 5 February 2012.
- ↑ Mac Lane and Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)
- ↑ King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
- ↑ Shafarevich (2003) says of a polynomial of degree zero, : "Such a polynomial is called a
*constant*because if we substitute different values of*x*in it, we always obtain the same value ." (p. 23) - ↑ James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (
*Mechanics Magazine*, Vol. LV, p. 171) - ↑ Lang, Sergei (2005).
*Algebra*(3rd ed.). Springer. p. 100. ISBN 978-0-387-95385-4. - ↑ Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27)

Childs (1995) uses −1. (p. 233)

Childs (2009) uses −∞ (p. 287), however he excludes zero polynomials in his Proposition 1 (p. 288) and then explains that the proposition holds for zero polynomials "with the reasonable assumption that +*m*= for*m*any integer or*m*= ".

Axler (1997) uses −∞. (p. 64)

Grillet (2007) says: "The degree of the zero polynomial 0 is sometimes left undefined or is variously defined as −1 ∈ ℤ or as , as long as deg 0 < deg A for all A ≠ 0." (A is a polynomial.) However, he excludes zero polynomials in his Proposition 5.3. (p. 121) - ↑ Barile, Margherita. "Zero Polynomial".
*MathWorld*. - ↑ 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)

In mathematics, a **finite field** or **Galois field** is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod *p* when *p* is a prime number.

In calculus, **Taylor's theorem** gives an approximation of a *k*-times differentiable function around a given point by a *k*-th order **Taylor polynomial**. For analytic functions the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought of as the extension of linear approximation to higher order polynomials, and in the case of *k* equals 2 is often referred to as a *quadratic approximation*. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial.

In algebra, the **discriminant** of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. For example, the discriminant of the quadratic polynomial

In mathematics, in particular abstract algebra, a **graded ring** is a ring that is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as **gradation** or **grading**.

In mathematics, a **quadric** or **quadric surface**, is a generalization of conic sections. It is a hypersurface in a (*D* + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in *D* + 1 variables. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a *degenerate quadric* or a *reducible quadric*.

In algebra, the **partial fraction decomposition** or **partial fraction expansion** of a rational function is an operation that consists of expressing the fraction as a sum of a polynomial and one or several fractions with a simpler denominator.

In mathematics, a **quadratic form** is a polynomial with terms all of degree two. For example,

In mathematics, an **affine algebraic plane curve** is the zero set of a polynomial in two variables. A **projective algebraic plane curve** is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation *h*(*x*, *y*, *t*) = 0 can be restricted to the affine algebraic plane curve of equation *h*(*x*, *y*, 1) = 0. These two operations are each inverse to the other; therefore, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In mathematics, a **rational function** is any function which can be defined by a **rational fraction**, *i.e.* an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field *K*. In this case, one speaks of a rational function and a rational fraction *over K*. The values of the variables may be taken in any field *L* containing *K*. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is *L*.

In number theory, the **local zeta function** is defined as

In mathematics, a **Bézout matrix** is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester (1853) and Arthur Cayley (1857) and named after Étienne Bézout. **Bézoutian** may also refer to the determinant of this matrix, which is equal to the resultant of the two polynomials. Bézout matrices are sometimes used to test the stability of a given polynomial.

In mathematics, the **resultant** of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root, or, equivalently, a common factor. In some older texts, the resultant is also called the **eliminant**.

**Polyharmonic splines** are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and and natural cubic splines in one dimension.

In number theory, an **average order of an arithmetic function** is some simpler or better-understood function which takes the same values "on average".

In algebraic geometry, a **morphism** between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a **regular map**. A morphism from an algebraic variety to the affine line is also called a **regular function**. A regular map whose inverse is also regular is called **biregular**, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well.

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.

A hyperelliptic curve is a class of algebraic curves. Hyperelliptic curves exist for every genus . The general formula of Hyperelliptic curve over a finite field is given by

In mathematics and optimization, a **pseudo-Boolean function** is a function of the form

In mathematics, the **ring of polynomial functions** on a vector space *V* over a field *k* gives a coordinate-free analog of a polynomial ring. It is denoted by *k*[*V*]. If *V* has finite dimension and is viewed as an algebraic variety, then *k*[*V*] is precisely the coordinate ring of *V*.

In mathematics a **P-recursive equation** can be solved for **polynomial solutions**. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of those recurrence equations with polynomial coefficients. The algorithm computes a *degree bound* for the solution in a first step. In a second step an ansatz for a polynomial of this degree is used and the unknown coefficients are computed by a system of linear equations. This article describes this algorithm.

- Axler, Sheldon (1997),
*Linear Algebra Done Right*(2nd ed.), Springer Science & Business Media - Childs, Lindsay N. (1995),
*A Concrete Introduction to Higher Algebra*(2nd ed.), Springer Science & Business Media - Childs, Lindsay N. (2009),
*A Concrete Introduction to Higher Algebra*(3rd ed.), Springer Science & Business Media - Grillet, Pierre Antoine (2007),
*Abstract Algebra*(2nd ed.), Springer Science & Business Media - King, R. Bruce (2009),
*Beyond the Quartic Equation*, Springer Science & Business Media - Mac Lane, Saunders; Birkhoff, Garrett (1999),
*Algebra*(3rd ed.), American Mathematical Society - Shafarevich, Igor R. (2003),
*Discourses on Algebra*, Springer Science & Business Media

- Polynomial Order; Wolfram MathWorld

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Images, videos and audio are available under their respective licenses.