## Degree of a Polynomial

In algebra degree of a polynomial is the maximum degree of the exponents of the variables of the monomials that compose it. Each degree has basically the same meaning when it refers to a polynomial or an algebraic equation. Consequently, the first definition that may need to be revised is that of the monomial, considered by elementary algebra as a basic algebraic expression, which is made up of a combination of numbers and letters (raised to positive integer exponents, including zero) between which there are no subtraction, addition or division operations, being then the only ones allowed, the multiplication raised between the numeric element (coefficient) and the non-numeric element (literal or variable) as well as the potentiation that occurred between the literal and its exponent.

The degree of a polynomial of a variable is the maximum exponent that the monomial has on the variable; For example, in 2 x 3 + 4 x 2 + x + 7, the term with the highest degree is 2 x 3 ; this term has a power of three on the variable x, and is therefore defined as degree 3 or third degree .

For polynomials of two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree of the polynomial will be the monomial of the highest degree. For example, the polynomial x 2 y 2 + 3 x 3 + 4 y has a degree of 4, the same degree as the term x 2 and 2 .

## Degree Of Congruence

Yes {\ displaystyle \ textstyle f (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + \ cdots + a_ {1} x ^ {1} + a_ {0 } x <0}{\ displaystyle \ textstyle f (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + \ cdots + a_ {1} x ^ {1} + a_ {0 } x <0} is a polynomial with integer coefficients, the degree of congruence of {\ displaystyle \ textstyle f (x) \ equiv 0 {\ pmod {m}}}{\ displaystyle \ textstyle f (x) \ equiv 0 {\ pmod {m}}}is the largest positive integer j such that a j is not congruent with zero with respect to modulus m .

**Equation Theory**

In algebraic equation theory , the degree of an equation corresponds to the maximum power to which the algebraic unknown in the equation is raised. For example:{\ displaystyle x ^ {3} y + 4x-y = 2xy \, \!}{\ displaystyle x ^ {3} y + 4x-y = 2xy \, \!}the equation is of the third degree in x , being of the first degree in the unknown y . See: Equation of second degree , Equation of third degree , Equation of fourth degree , Equation of fifth degree , etc.

**Degree of an extension**

In algebra it has the extension body and the degree therein is defined as any vector space with basis you can calculate the dimension of{\ displaystyle L}L as vector space over {\ displaystyle K}K, denoted by {\ displaystyle \ dim _ {K} (L)}{\ displaystyle \ dim _ {K} (L)}. {\ displaystyle L: K}{\ displaystyle L: K} to the dimension of {\ displaystyle L}L What {\ displaystyle K}K-vectorial space: {\ displaystyle [L: K] = \ dim _ {K} (L)}{\ displaystyle [L: K] = \ dim _ {K} (L)}.

**Graph theory**

In Graph Theory , the degree or valence of a vertex is the number of edges incident to the vertex. The degree of a vertex x is denoted by degree (x) , g (x) or gr (x) (although δ (x) is also used , and from English d (x) and deg (x) ). The maximum degree of a graph G is denoted by Δ (G) and the minimum degree of a graph G is denoted by δ (G) .

**Degree of freedom (statistics)**

In statistics , degrees of freedom , an expression introduced by Ronald Fisher , says that, from a set of observations, the degrees of freedom are given by the number of values that can be arbitrarily assigned, before the rest of the variables take a value automatically, the product of establishing those that are free; this, in order to compensate and equalize a result which has been previously known. They are found by the formula{\ displaystyle nr}{\ displaystyle nr}, where n is the number of subjects in the sample that can take a value and r is the number of subjects whose value will depend on the number taken by the members of the sample who are free. They can also be represented by{\ displaystyle kr}{\ displaystyle kr}, where {\ displaystyle k}k= number of groups; this, when operations are carried out with groups and not with individual subjects.

When it comes to eliminating statistics with a set of data, the residuals – expressed in vector form – are found, usually in a space of less dimension than that in which the original data were found. The degrees of freedom of the error are determined precisely by the value of this smallest dimension.

This also means that the residuals are restricted to being in a space of dimension {\ displaystyle n-1}n-1 (in this example, in the general case a {\ displaystyle nr}{\ displaystyle nr}) since, if the value of {\ displaystyle n-1}n-1of these residues, the determination of the value of the remaining residue is immediate. Thus, it is said that «the error has{\ displaystyle n-1}n-1 degrees of freedom »(the error has {\ displaystyle nr}{\ displaystyle nr} general degrees of freedom).