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 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 , 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, , the degree of the polynomial is 3.
The following names are assigned to polynomials according to their degree:
The canonical forms of the three examples above are:
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 the product of two polynomials is the sum of their degrees
It is convenient, however, to define that the degree of the zero polynomial is minus infinity, −∞, and introduce the rules
The price to be paid for saving the rules for computing the degree of sums and products of polynomials is that the general rule
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.
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:
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).
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.
US Patent Issued to LSI on Feb. 19 for "Reducing a Degree of a Polynomial in a Polynomial Division Calculation" (Israeli Inventors)
Feb 20, 2013; ALEXANDRIA, Va., Feb. 20 -- United States Patent no. 8,381,080, issued on Feb. 19, was assigned to LSI Corp. (San Jose,...
WIPO ASSIGNS PATENT TO AMPERSAND INTERNATIONAL FOR "METHOD FOR NOISE FILTERING BASED ON CONFIDENCE INTERVAL EVALUATION" (RUSSIAN INVENTORS)
Sep 01, 2011; GENEVA, Sept. 1 -- Publication No. WO/2011/106527 was published on Sept. 01. Title of the invention: "METHOD FOR NOISE FILTERING...