The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. For example, in 2x³ + 4x² + x + 7, the term of highest degree is 2x³; therefore the polynomial is said to have degree 3. Sometimes the same concept is called the order of the polynomial. In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

Examples

• The polynomial 3 − 5x + 2x⁵ − 7x⁹ has degree 9.
• The polynomial (y − 3)(2y + 6)( − 4y − 21) has degree 3.
• The polynomial (3z⁸ + z⁵ − 4z² + 6) + ( − 3z⁸ + 8z⁴ + 2z³ + 14z) has degree 5.

In general, to determine the degree of a polynomial expression, the expression has to be brought in "canonical form" by multiplying out until all terms are a product of constants and variables, in which terms with the same product of variables are collected together, and terms in which the constant factor is zero are elided. Usually (but not necessarily) the terms are also ordered from highest to lowest degree. The canonical forms of the three examples above are:

• for 3 − 5x + 2x⁵ − 7x⁹, after reordering, − 7x⁹ + 2x⁵ − 5x + 3;
• for (y − 3)(2y + 6)( − 4y − 21), after multiplying out and collecting terms of the same degree, − 8y³ − 42y² + 72y + 378;
• for (3z⁸ + z⁵ − 4z² + 6) + ( − 3z⁸ + 8z⁴ + 2z³ + 14z), in which the two terms of degree 8 cancel, z⁵ + 8z⁴ + 2z³ − 4z² + 14z + 6.

Behaviour under addition, subtraction and multiplication

The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees i.e.

.

.

For example:

• The degree of (x³ + x) + (x² + 1) = x³ + x² + x + 1 is 3. Note that 3 ≤ max(3,2)
• The degree of (x³ + x) − (x³ + x²) = − x² + x is 2. Note that 2 ≤ max(3,3)

The degree of the product of two polynomials is the sum of their degrees

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

For example:

• The degree of (x³ + x)(x² + 1) = x⁵ + 2x³ + x is 3+2 = 5.

The degree of the zero polynomial

The function f(x)=0 is a polynomial, called the zero polynomial. It has no terms, and so, strictly speaking, it has no degree either. The above rules for the degree of sums and products of polynomials do not apply if any of the polynomials involved is the zero polynomial.

It is convenient, however, to define that the degree of the zero polynomial is minus infinity, −∞, and introduce the rules

,

and

.

For example:

• The degree of the sum is 3. Note that .
• The degree of the difference is . Note that .
• The degree of the product is .

The price to be paid for saving the rules for computing the degree of sums and products of polynomials is that the general rule

,

breaks down when .

The degree computed from the function values

If a polynomial f(x) has positive values for sufficiently large values of x, then the degree of that polynomial can be computed by the formula

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 ∞.

The reciprocal function: y = 1/x. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... Above is the graph plots of Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... The exponential function is one of the most important functions in mathematics. ...

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 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.

x²y² + 3x³ + 4y = (3)x³ + (y²)x² + (4y) = (x²)y² + (4)y + (3x³)

This 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. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...

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 (mod 4) = 0. Therefore, let f(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 of f and g (which each had degree 1). Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Look up field in Wiktionary, the free dictionary A green field or paddock Field may refer to: A field is an open land area, used for growing agricultural crops. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

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.

Names of polynomials by degree

Polynomials with small degrees may be named according to their degree as follows:

• Degree 1 - linear
• Degree 2 - quadratic
• Degree 3 - cubic
• Degree 4 - quartic
• Degree 5 - quintic
• Degree 6 - sextic or hexic
• Degree 7 - septic or heptic
• Degree 8 - octic
• Degree 9 - nonic
• Degree 10 - decic
• Degree 100 - hectic^[1]

The names beyond "quintic" are uncommon and rarely used. The word linear comes from the Latin word linearis, which means created by lines. ... f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ... Polynomial of degree 4: f(x) = (x+4)(x+1)(x-1)(x-3)/14+0. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ...

See also

• Degree (mathematics) — other meanings of degree in mathematics

This article is about the term degree as used in mathematics. ...

References

1. ^ Miami Dade College; Polynomials