Degree of a polynomial

Names of polynomials by degree

The following names are assigned to polynomials according to their degree:

Special case – zero (see § Degree of the zero polynomial below)

Degree 0 – constant

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)

For higher degrees, names have sometimes been proposed, but they are rarely used:

Degree 8 – octic

Degree 9 – nonic

Degree 10 – decic

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 x 2 + x y + y 2 , is called a "binary quadratic": binary due to two variables, quadratic due to degree two. 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 x 2 + y 2 is a "binary quadratic binomial".

Other examples

The polynomial 3 − 5 x + 2 x 5 − 7 x 9 is a nonic polynomial

The polynomial ( y − 3 ) ( 2 y + 6 ) ( − 4 y − 21 ) is a cubic polynomial

The polynomial ( 3 z 8 + z 5 − 4 z 2 + 6 ) + ( − 3 z 8 + 8 z 4 + 2 z 3 + 14 z ) is a quintic polynomial (as the z 8 are cancelled out)

The canonical forms of the three examples above are:

for 3 − 5 x + 2 x 5 − 7 x 9 , after reordering, − 7 x 9 + 2 x 5 − 5 x + 3 ;

for ( y − 3 ) ( 2 y + 6 ) ( − 4 y − 21 ) , after multiplying out and collecting terms of the same degree, − 8 y 3 − 42 y 2 + 72 y + 378 ;

for ( 3 z 8 + z 5 − 4 z 2 + 6 ) + ( − 3 z 8 + 8 z 4 + 2 z 3 + 14 z ) , in which the two terms of degree 8 cancel, z 5 + 8 z 4 + 2 z 3 − 4 z 2 + 14 z + 6 .

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.

Behaviour under 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.

deg ⁡ ( P + Q ) ≤ max ( deg ⁡ ( P ) , deg ⁡ ( Q ) ) . deg ⁡ ( P − Q ) ≤ max ( deg ⁡ ( P ) , deg ⁡ ( Q ) ) .

E.g.

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

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

Behaviour under 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.

deg ⁡ ( c P ) = deg ⁡ ( P ) .

E.g.

The degree of 2 ( x 2 + 3 x − 2 ) = 2 x 2 + 6 x − 4 is 2, just as the degree of x 2 + 3 x − 2 .

Note that for polynomials over a ring containing divisors of zero, this is not necessarily true. For example, in Z / 4 Z , deg ⁡ ( 1 + 2 x ) = 1 , but deg ⁡ ( 2 ( 1 + 2 x ) ) = deg ⁡ ( 2 + 4 x ) = deg ⁡ ( 2 ) = 0 .

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.)

Behaviour under multiplication

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

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

E.g.

The degree of ( x 3 + x ) ( x 2 + 1 ) = x 5 + 2 x 3 + x is 3 + 2 = 5.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in Z / 4 Z , deg ⁡ ( 2 x ) + deg ⁡ ( 1 + 2 x ) = 1 + 1 = 2 , but deg ⁡ ( 2 x ( 1 + 2 x ) ) = deg ⁡ ( 2 x ) = 1 .

Behaviour under composition

The degree of the composition of two non-constant polynomials P and Q over a field or integral domain is the product of their degrees:

deg ⁡ ( P ∘ Q ) = deg ⁡ ( P ) deg ⁡ ( Q ) .

E.g.

If P = ( x 3 + x ) , Q = ( x 2 + 1 ) , then P ∘ Q = P ∘ ( x 2 + 1 ) = ( x 2 + 1 ) 3 + ( x 2 + 1 ) = x 6 + 3 x 4 + 4 x 2 + 2 , which has degree 6.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in Z / 4 Z , deg ⁡ ( 2 x ) deg ⁡ ( 1 + 2 x ) = 1 ⋅ 1 = 1 , but deg ⁡ ( 2 x ∘ ( 1 + 2 x ) ) = deg ⁡ ( 2 + 4 x ) = deg ⁡ ( 2 ) = 0 .

Degree of the zero polynomial

The degree of the zero polynomial is either left undefined, or is defined to be negative (usually −1 or −∞).

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.

It is convenient, however, to define the degree of the zero polynomial to be negative infinity, −∞, and introduce the arithmetic rules

max ( a , − ∞ ) = a ,

and

a + − ∞ = − ∞ .

These examples illustrate how this extension satisfies the behavior 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 ≤ max ( 3 , − ∞ ) .

The degree of the difference ( x ) − ( x ) = 0 is − ∞ . This satisfies the expected behavior, which is that − ∞ ≤ max ( 1 , 1 ) .

The degree of the product ( 0 ) ( x 2 + 1 ) = 0 is − ∞ . This satisfies the expected behavior, which is that − ∞ = − ∞ + 2 .

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

deg ⁡ f = lim x → ∞ log ⁡ | f ( x ) | log ⁡ x ;

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:

The degree of the multiplicative inverse,   1 / x , is −1.

The degree of the square root, x , is 1/2.

The degree of the logarithm,   log ⁡ x , is 0.

The degree of the exponential function,   exp ⁡ x , is ∞.

Note that the formula also gives sensible results for many combinations of such functions, e.g., the degree of 1 + x x is − 1 / 2 .

Another formula to compute the degree of f from its values is

deg ⁡ f = lim x → ∞ x f ′ ( x ) f ( x ) ;

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 d x 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 using big O notation. In the analysis of algorithms, it is for example often relevant to distinguish between the growth rates of x and x log ⁡ x , 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.

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.

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 = Z / 4 Z , 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(f⋅g) = 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.