Polynomial Addition and Multiplication

Kenneth Levasseur

Mathematical Sciences

UMass Lowell

Kenneth_Levasseur@uml.edu

Calculations done here use AbstractAlgebra a Mathematica package available at http://www.central.edu/eaam/index.asp

You add and multiply polynomials over an arbitrary ring the same way you learned to do it for polynomials over the real number in high school algebra. The only thing you have to keep in mind is that the coefficient operations use the underlying operations for the ring over which you are working. For example, if you working with polynomials over then adding and involved adding "like terms" and the result is

Addition

Given two polynomials

Their sum is the polynomial

You just add corresponding coefficients to add. Notice that we didn't specify exactly where either two of and "stop" but eventually the coefficients of both are zero and the number of additions in the underlying ring is no more than the largest degree of the two addends.

Example

Multiplication

Given two polynomials of degree m and n, respectively

Their product is the polynomials is computed by multiplying all of the terms in the first polynomial by each term in the second. This involves multiplying powers of the indeterminate x, which follows the rule , where the addition is of integers, not in the underlying ring. If we organize our results the product follows a regular pattern, but somewhat more complicated than for addition.

In both of the formulae above it should be understood that coefficients with index higher than the respective degrees are equal to the zero of the underlying ring. Notice that for each term in the product the coefficient of is the sum of products of coefficients whose indices add up to k.

In multiplying polynomials of degree m and n, the number of multiplications involving nonzero coefficients can be as high as

Example

This product isn't as easy to see as the sum above. Work out a few coefficients to check the result.

Here's why we will usually work with polynomials over a field. The following strange things don't happen if a ring doesn't have zero divisors as has.

A zero divisor among the polynomials

Here we see that t is a unit, which isn't bad; but normally we expect the degree of a product to be the sum of degrees, but not in this polynomial ring.

Over a field this strange result never occurs the product of two linear polynomials is always a quadratic polynomial.

Created by Mathematica (September 13, 2005)