Definitions

# Degree of a polynomial

When a polynomial is expressed as a sum or difference of terms (e.g., in standard or canonical form), the exponent of the term with the highest exponent is the degree of the polynomial. The degree of a term is the sum of the powers of each variable in the term. The words degree and order are used interchangeably.

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 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 $\left(y - 3\right)\left(2y + 6\right)\left(-4y - 21\right)$, first express the polynomial as a sum or difference of terms by multiplying each of its factors, combine any like terms, then determine its degree. Since, $\left(y - 3\right)\left(2y + 6\right)\left(-4y - 21\right) = -8y^3 - 42y^2 + 72y + 378$, the degree of the polynomial is 3.

## Names of polynomials by degree

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

et cetera.

## Other Examples

• The polynomial $3 - 5 x + 2 x^5 - 7 x^9$ has degree 9.
• The polynomial $\left(y - 3\right)\left(2y + 6\right)\left(-4y - 21\right)$ has degree 3.
• The polynomial $\left(3 z^8 + z^5 - 4 z^2 + 6\right) + \left(-3 z^8 + 8 z^4 + 2 z^3 + 14 z\right)$ has degree 5.

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 $\left(y - 3\right)\left(2y + 6\right)\left(-4y - 21\right)$, after multiplying out and collecting terms of the same degree, $- 8 y^3 - 42 y^2 + 72 y + 378$;
• for $\left(3 z^8 + z^5 - 4 z^2 + 6\right) + \left(-3 z^8 + 8 z^4 + 2 z^3 + 14 z\right)$, 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 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.

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

For example:

• The degree of $\left(x^3+x\right)+\left(x^2+1\right)=x^3+x^2+x+1$ is 3. Note that 3 ≤ max(3,2)
• The degree of $\left(x^3+x\right)-\left(x^3+x^2\right)=-x^2+x$ is 2. Note that 2 ≤ max(3,3)

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

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

For example:

• The degree of $\left(x^3+x\right)\left(x^2+1\right)=x^5+2x^3+x$ 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

$max\left(a,-infty\right) = a$,
and
$a + -infty = -infty$.

For example:

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

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
$deg f = lim_\left\{xrarrinfty\right\}frac\left\{log |f\left(x\right)|\right\}\left\{log x\right\}.$
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 ∞.

## 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.

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