Polynomial arithmetic
Encyclopedia
Polynomial arithmetic includes basic mathematical operations such as addition
, subtraction
, and multiplication
. These operations are defined naturally as if the variable
was an element of 
. Division
is defined similarly, but requires that
be a field
. Examples of fields include rational numbers,
for 
prime
, and real numbers. The set of all integers is not a field and does not support polynomial division.
then addition is defined as
then multiplication is defined as 
where 
. Note that we treat 
as zero for 
and that the degree of the product is equal to the sum
of the degrees to the two polynomials.
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, and multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
. These operations are defined naturally as if the variable
Variable (mathematics)
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...
was an element of . DivisionDivision (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
is defined similarly, but requires that
be a fieldField (mathematics)
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...
. Examples of fields include rational numbers,
for primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, and real numbers. The set of all integers is not a field and does not support polynomial division.
Addition and subtraction
Addition and subtraction are performed by adding or subtracting corresponding coefficients. If
then addition is defined as
-
where m > n
Multiplication
Multiplication is performed much the same way as addition and subtraction, but instead by multiplying the corresponding coefficients. If then multiplication is defined as where . Note that we treat as zero for and that the degree of the product is equal to the sumSUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of the degrees to the two polynomials.