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.
Addition and subtraction are performed by adding or subtracting corresponding coefficients. If
then addition is defined as
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 sum of the degrees to the two polynomials.