The degree of a polynomial
represents the highest degree of a polynominal's terms
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...
(with non-zero coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
), should the polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
be expressed in canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
(i.e. as a sum or difference of terms). The degree of an individual term is the sum of the exponents acting on the term's variables
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. The word degree
has been favored for some decades in standard textbooks - but in some older books, the word order
may be used instead.
For example, the polynomial
has three terms. (Notice, this polynomial can also be expressed as
.) The first term has a degree of 5 (the sum of the powers
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
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 (for example
) it is easier to expand or express the polynomial into a sum or difference of terms; this may be achieved by multiplying each of its factors
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
, and combining monomial terms. This makes the exponents more obvious, and easier to determine when calculating the degree of the equation. Since,
, the degree of the polynomial can be found to be 3.
Names of polynomials by degree
The following names are assigned to polynomials according to their degree:
- Degree 0 – constant
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...
- Degree 1 – linear
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
- Degree 2 – quadratic
In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2...
- Degree 3 – cubic
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
- Degree 4 – quartic
In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...
(or, less commonly, biquadratic) (or, a little more common, Fourth degree)
- Degree 5 – quintic
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
- Degree 6 – sextic
In mathematics, a sextic equation is a polynomial equation of degree six. It is of the form:ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\,where a \neq 0....
(or, less commonly, hexic)
- Degree 7 – septic
In mathematics, a septic equation, heptic equation or septimic equation is a polynomial equation of degree seven. It is of the form:ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,...
(or, less commonly, heptic)
- Degree 8 – octic
- Degree 9 – nonic
- Degree 10 – decic
- Degree 100 - hectic
The degree of the zero polynomial is either left explicitly undefined, or is defined to be negative (usually −1 or −∞).
- The polynomial is a nonic polynomial
- The polynomial is a cubic polynomial
- The polynomial is a quintic polynomial.
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, .
Behavior under addition, subtraction, multiplication and function composition
The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees i.e.
- 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 two polynomials is the sum of their degrees
- The degree of is 3 + 2 = 5.
The degree of the composition of two polynomials is the product of their degrees
- If , , then , which has degree 6.
The degree of the zero polynomial
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. 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 the degree of the zero polynomial to be minus infinity
, −∞, and introduce the rules
- 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
The degree of a polynomial f
can be computed by the formula
This formula generalizes the concept of degree to some functions that are not polynomials.
- The degree of the multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
, , is −1.
- The degree of the square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
, , is 1/2.
- The degree of the logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
, , is 0.
- The degree of the exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
, , is ∞.
Another formula to compute the degree of f
from its values is
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
has degree 4, the same degree as the term x2y2
However, a polynomial in variables x
, 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
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R, the polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
] is the set of all polynomials in x
that have coefficients chosen from R. In the special case that R is also a field
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, then the polynomial ring R[x
] is a principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
and, more importantly to our discussion here, a Euclidean domain
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
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
) and g
), the degree of the product f
) must be larger than both the degrees of f
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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
4. This ring is not a field (and is not even an integral domain
) because 2•2 = 4 (mod 4) = 0. Therefore, let f
) = g
) = 2x
+ 1. Then, f
) = 4x2
+ 1 = 1. Thus deg(f
) = 0 which is not greater than the degrees of f
(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
) = 0 to also be undefined so that it follows the rules of a norm in a euclidean domain.