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Degree of a polynomial

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Mathematical concept

Inmathematics, thedegree of apolynomial is the highest of the degrees of the polynomial'smonomials (individual terms) with non-zero coefficients. Thedegree of a term is the sum of the exponents of thevariables that appear in it, and thus is a non-negativeinteger. For aunivariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.^[1] The termorder has been used as a synonym ofdegree but, nowadays, may refer to several other concepts (seeOrder of a polynomial (disambiguation)).

For example, the polynomial $7x^{2}y^{3}+4x-9,$ which can also be written as $7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},$ has three terms. The first term has a degree of 5 (the sum of thepowers 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 $(x+1)^{2}-(x-1)^{2}$ , one can put it in standard form by expanding the products (bydistributivity) and combining the like terms; for example, $(x+1)^{2}-(x-1)^{2}=4x$ 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

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Look upAppendix:English polynomial degrees in Wiktionary, the free dictionary.

The following names are assigned to polynomials according to their degree:^[2]^[3]^[4]

Special case –zero (see§ Degree of the zero polynomial, below)
Degree 0 – non-zeroconstant^[5]
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)
Degree 8 – octic
Degree 9 – nonic
Degree 10 – decic

Names for degree above three are based on Latinordinal numbers, and end in-ic. This should be distinguished from the names used for the number of variables, thearity, which are based on Latindistributive numbers, and end in-ary. For example, a degree two polynomial in two variables, such as $x^{2}+xy+y^{2}$ , 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 aremonomial,binomial, and (less commonly)trinomial; thus $x^{2}+y^{2}$ is a "binary quadratic binomial".

Examples

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The polynomial $(y-3)(2y+6)(-4y-21)$ is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes $-8y^{3}-42y^{2}+72y+378$ , with highest exponent 3.

The polynomial $(3z^{8}+z^{5}-4z^{2}+6)+(-3z^{8}+8z^{4}+2z^{3}+14z)$ is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving $z^{5}+8z^{4}+2z^{3}-4z^{2}+14z+6$ , with highest exponent 5.

Behavior under polynomial operations

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

Addition

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The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; that is,

\deg(P+Q)\leq \max\{\deg(P),\deg(Q)\}

and

\deg(P-Q)\leq \max\{\deg(P),\deg(Q)\}

For example, the degree of $(x^{3}+x)-(x^{3}+x^{2})=-x^{2}+x$ is 2, and 2 ≤ max{3, 3}.

The equality always holds when the degrees of the polynomials are different. For example, the degree of $(x^{3}+x)+(x^{2}+1)=x^{3}+x^{2}+x+1$ is 3, and 3 = max{3, 2}.

Multiplication

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The degree of the product of a polynomial by a non-zeroscalar is equal to the degree of the polynomial; that is,

\deg(cP)=\deg(P)

For example, the degree of $2(x^{2}+3x-2)=2x^{2}+6x-4$ is 2, which is equal to the degree of $x^{2}+3x-2$ .

Thus, theset of polynomials (with coefficients from a given fieldF) whose degrees are smaller than or equal to a given numbern forms avector space; for more, seeExamples of vector spaces.

More generally, the degree of the product of two polynomials over afield or anintegral domain is the sum of their degrees:

\deg(PQ)=\deg(P)+\deg(Q)

For example, the degree of $(x^{3}+x)(x^{2}+1)=x^{5}+2x^{3}+x$ is 5 = 3 + 2.

For polynomials over an arbitraryring, the above rules may not be valid, because of cancellation that can occur when multiplying two nonzero constants. For example, in the ring $\mathbf {Z} /4\mathbf {Z}$ ofintegers modulo 4, one has that $\deg(2x)=\deg(1+2x)=1$ , but $\deg(2x(1+2x))=\deg(2x)=1$ , which is not equal to the sum of the degrees of the factors.

Composition

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The degree of the composition of two non-constant polynomials $P {\displaystyle P}$ and $Q {\displaystyle Q}$ over a field or integral domain is the product of their degrees: $\deg(P\circ Q)=\deg(P)\deg(Q).$

For example, if $P=x^{3}+x$ has degree 3 and $Q=x^{2}-1$ has degree 2, then their composition is $P\circ Q=P\circ (x^{2}-1)=(x^{2}-1)^{3}+(x^{2}-1)=x^{6}-3x^{4}+4x^{2}-2,$ which has degree 6.

Note that for polynomials over an arbitrary ring, the degree of the composition may be less than the product of the degrees. For example, in $\mathbf {Z} /4\mathbf {Z} ,$ the composition of the polynomials $2x$ and $1+2x$ (both of degree 1) is the constant polynomial $2x\circ (1+2x)=2+4x=2,$ of degree 0.

Degree of the zero polynomial

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The degree of thezero polynomial is either left undefined, or is defined to be negative (usually −1 or $-\infty$ ).^[7]

Like any constant value, the value 0 can be considered as a (constant) polynomial, called thezero polynomial. 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.^[8]

It is convenient, however, to define the degree of the zero polynomial to benegative infinity, $-\infty ,$ and to introduce the arithmetic rules^[9]

\max(a,-\infty )=a,

and

a+(-\infty )=-\infty .

These examples illustrate how this extension satisfies thebehavior rules above:

The degree of the sum $(x^{3}+x)+(0)=x^{3}+x$ is 3. This satisfies the expected behavior, which is that $3\leq \max(3,-\infty )$ .
The degree of the difference $(x)-(x)=0$ is $-\infty$ . This satisfies the expected behavior, which is that $-\infty \leq \max(1,1)$ .
The degree of the product $(0)(x^{2}+1)=0$ is $-\infty$ . This satisfies the expected behavior, which is that $-\infty =-\infty +2$ .

Computed from the function values

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A number of formulae exist which will evaluate the degree of a polynomial functionf. One based onasymptotic analysis is

\deg f=\lim _{x\rightarrow \infty }{\frac {\log |f(x)|}{\log x}}

;

this is the exact counterpart of the method of estimating the slope in alog–log plot.

This formula generalizes the concept of degree to some functions that are not polynomials.For example:

The degree of themultiplicative inverse, $\ 1/x$ , is −1.
The degree of thesquare root, ${\sqrt {x}}$ , is 1/2.
The degree of thelogarithm, $\ \log x$ , is 0.
The degree of theexponential function, $\exp x$ , is $\infty .$

The formula also gives sensible results for many combinations of such functions, e.g., the degree of ${\frac {1+{\sqrt {x}}}{x}}$ is $-1/2$ .

Another formula to compute the degree off from its values is

\deg f=\lim _{x\to \infty }{\frac {xf'(x)}{f(x)}}

;

this second formula follows from applyingL'Hôpital's rule to the first formula. Intuitively though, it is more about exhibiting the degreed as the extra constant factor in thederivative $dx^{d-1}$ of $x^{d}$ .

A more fine grained (than a simple numeric degree) description of the asymptotics of a function can be had by usingbig O notation. In theanalysis of algorithms, it is for example often relevant to distinguish between the growth rates of $x {\displaystyle x}$ and $x\log x$ , which would both come out as having thesame degree according to the above formulae.

Extension to polynomials with two or more variables

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For polynomials in two or more variables, the degree of a term is thesum of the exponents of the variables in the term; the degree (sometimes called thetotal degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomialx²y² + 3x³ + 4y has degree 4, the same degree as the termx²y².

However, a polynomial in variablesx andy, is a polynomial inx with coefficients which are polynomials iny, and also a polynomial iny with coefficients which are polynomials inx. The polynomial

x^{2}y^{2}+3x^{3}+4y=(3)x^{3}+(y^{2})x^{2}+(4y)=(x^{2})y^{2}+(4)y+(3x^{3})

has degree 3 inx and degree 2 iny.

Degree function in abstract algebra

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Given aringR, thepolynomial ringR[x] is the set of all polynomials inx that have coefficients inR. In the special case thatR is also afield, the polynomial ringR[x] is aprincipal ideal domain and, more importantly to our discussion here, aEuclidean domain.

It can be shown that the degree of a polynomial over a field satisfies all of the requirements of thenorm function in the euclidean domain. That is, given two polynomialsf(x) andg(x), the degree of the productf(x)g(x) must be larger than both the degrees off andg 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. LetR = $\mathbb {Z} /4\mathbb {Z}$ , the ring of integersmodulo 4. This ring is not a field (and is not even anintegral domain) because 2 × 2 = 4 ≡ 0 (mod 4). Therefore, letf(x) =g(x) = 2x + 1. Then,f(x)g(x) = 4x² + 4x + 1 = 1. Thus deg(f⋅g) = 0 which is not greater than the degrees off andg (which each had degree 1).

Since thenorm function is not defined for the zero element of the ring, we consider the degree of the polynomialf(x) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.

Notes

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^For simplicity, this is ahomogeneous polynomial, with equal degree in both variables separately.

^Gullberg, Jan (1997),Mathematics From the Birth of Numbers, W. W. Norton & Company, p. 128,ISBN 9780393040029
^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".
^James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (Mechanics Magazine, Vol. LV, p. 171)
^Shafarevich (2003) says of a polynomial of degree zero, $f(x)=a_{0}$ : "Such a polynomial is called aconstant because if we substitute different values ofx in it, we always obtain the same value $a_{0}$ ." (p. 23)
^Lang, Serge (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 $-\infty$ +m = $-\infty$ form any integer orm = $-\infty$ ".
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 ∈ $\mathbb {Z}$ or as $-\infty$ , as long as deg 0 < degA for allA ≠ 0." (A is a polynomial.) However, he excludes zero polynomials in his Proposition 5.3. (p. 121)
^Caldwell, William (2009), "Applying Concept Mapping to Algebra I", in Afamasaga-Fuata'i, Karoline (ed.),Concept Mapping in Mathematics: Research into Practice, Springer, pp. 217–234,doi:10.1007/978-0-387-89194-1_11,ISBN 9780387891941; see section "Degree of a polynomial", pp. 225–226: "The product of the zero polynomial [with] any other polynomial is always the zero polynomial, so such a property of degrees (the degree of the product is the sum of the degrees of the two factors) would not hold if one of the two polynomials were the polynomial 0. That is why we do not assign a degree to the zero polynomial."
^Axler (1997) gives these rules and says: "The 0 polynomial is declared to have degree $-\infty$ so that exceptions are not needed for various reasonable results." (p. 64)

References

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Axler, Sheldon (1997),Linear Algebra Done Right (2nd ed.), Springer Science & Business Media,ISBN 9780387982595
Childs, Lindsay N. (1995),A Concrete Introduction to Higher Algebra (2nd ed.), Springer Science & Business Media,ISBN 9780387989990
Childs, Lindsay N. (2009),A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media,ISBN 9780387745275
Grillet, Pierre Antoine (2007),Abstract Algebra (2nd ed.), Springer Science & Business Media,ISBN 9780387715681
King, R. Bruce (2009),Beyond the Quartic Equation, Springer Science & Business Media,ISBN 9780817648497
Mac Lane, Saunders;Birkhoff, Garrett (1999),Algebra (3rd ed.), American Mathematical Society,ISBN 9780821816462
Shafarevich, Igor R. (2003),Discourses on Algebra, Springer Science & Business Media,ISBN 9783540422532

v t e Polynomials andpolynomial functions
Bydegree	Zero polynomial (degree undefined or −1 or −∞) Constant function (0) Linear function (1) Linear equation Quadratic function (2) Quadratic equation Cubic function (3) Cubic equation Quartic function (4) Quartic equation Quintic function (5) Sextic equation (6) Septic equation (7)
By properties	Univariate Bivariate Multivariate Monomial Binomial Trinomial Irreducible Square-free Homogeneous Quasi-homogeneous
Tools and algorithms	Factorization Greatest common divisor Division Horner's method of evaluation Polynomial identity testing Resultant Discriminant Gröbner basis