Degree of a polynomial

The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e., as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The term order refers to a different, but related concept.

For example, the polynomial $7x^2y^3 + 4x - 9$ has three terms. (Notice, this polynomial can also be expressed as $7x^2y^3 + 4x^1y^0 - 9x^0y^0$.) 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.

To determine the degree of a polynomial that is not in standard form (for example $(x+1)^2-(x-1)^2$), one has to put it first in standard form by expanding the products (by distributivity) and combining the like terms; for example $(x+1)^2-(x-1)^2= 4x$, and its degree is 1, although 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]

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

Other examples

• The polynomial $3 - 5 x + 2 x^5 - 7 x^9$ is a nonic polynomial
• The polynomial $(y - 3)(2y + 6)(-4y - 21)$ 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 $(y - 3)(2y + 6)(-4y - 21)$, 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 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.

$\deg(P + Q) \leq \max(\deg(P),\deg(Q))$.
$\deg(P - Q) \leq \max(\deg(P),\deg(Q))$.

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)

The degree of the product of a polynomial by a non-zero scalar is equal to the degree of the polynomial, i.e.

$\deg(cP)=\deg(P)$.

E.g.

• The degree of $2(x^2+3x-2)=2x^2+6x-4$ is 2, just as the degree of $x^2+3x-2$.

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

The collection 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 collection is not a ring, as it is not closed under multiplication, as is seen below.)

The degree of the product of two polynomials over a field is the sum of their degrees

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

E.g.

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

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

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

$\deg(P \circ Q) = \deg(P)\deg(Q)$.

E.g.

• If $P = (x^3+x)$, $Q = (x^2+1)$, then $P \circ Q = P \circ (x^2+1) = (x^2+1)^3+(x^2+1) = x^6+3x^4+4x^2+2$, which has degree 6.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in $\mathbf{Z}/4\mathbf{Z}$, $\deg(2x) \deg(1+2x) = 1\cdot 1 = 1$, but $\deg(2x\circ(1+2x)) = \deg(2+4x)=\deg(2) = 0$.

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. As such, the degree of any such function is undefined. 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.[2]

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

$\max(a,-\infty) = a, \,$

and

$a + -\infty = -\infty. \,$

For example:

• The degree of the sum $\ (x^3+x)+(0)=x^3+x$ is 3. Note that $3 \le \max(3, -\infty)$.
• The degree of the difference $\ x-x = 0$ is $-\infty$. Note that $\ -\infty \le \max(1,1)$.
• The degree of the product $\ (0)(x^2+1)=0$ is $\ (-\infty)+2 = -\infty$.

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

$\ a+b=a \quad \Rightarrow \quad b=0, \,$

breaks down when $\ a = -\infty$.

The degree computed from the function values

The degree of a polynomial f can be computed by the formula

$\deg f = \lim_{x\rarr\infty}\frac{\log |f(x)|}{\log x}.$

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, $\sqrt x$, is 1/2.
• The degree of the logarithm, $\ \log x$, is 0.
• The degree of the exponential function, $\ \exp x$, is ∞.

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

$\deg f = \lim_{x\to\infty}\frac{x f'(x)}{f(x)}.$

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 = $\mathbb{Z}/4\mathbb{Z}$, 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(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.