Degree of a polynomial

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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.

Polynomial In mathematics, sum of products of variables, power of variables, and coefficients

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 x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 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.

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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.

Names of polynomials by degree

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

Constant function mathematical function whose (output) value is the same for every input value

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:

Cubic function a polynomial function in a single variable where the highest exponent of the variable is 3

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:

Octic equation

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".

Other examples

The canonical forms of the three examples above are:

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. [6]

Addition

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.

Scalar multiplication

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.

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.)

Multiplication

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

.

E.g.

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

Composition

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.

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 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:

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.

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

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.

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 x2y2 + 3x3 + 4y has degree 4, the same degree as the term x2y2.

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 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) = 2x + 1. Then, f(x)g(x) = 4x2 + 4x + 1 = 1. Thus deg(fg) = 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.

Notes

  1. For simplicity, this is a homogeneous polynomial, with equal degree in both variables separately.
  1. "Names of Polynomials". 25 November 1997. Retrieved 5 February 2012.
  2. Mac Lane and Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)
  3. King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
  4. 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)
  5. James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (Mechanics Magazine, Vol. LV, p. 171)
  6. Lang, Sergei (2005). Algebra (3rd ed.). Springer. p. 100. ISBN   978-0-387-95385-4.
  7. 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)
  8. Barile, Margherita. "Zero Polynomial". MathWorld .
  9. 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)

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