One Reason We Study Polynomials

It has been said that 90% of all Mathematics relies on the ability to recognize patterns.  In addition to applications that involve polynomials, such as calculating postage, the patterns of polynomials appear in other places.  One of the places that polynomial patterns appear is in the numbers we use everyday.  It is for this reason that polynomials have been frequently used to “model” and study numbers.  Let’s see how it is that this pattern appears.

First, let us take a specific number to work with, say 2,457.  Now, write the number in expanded form

Separate each term into components that might be used if you were counting out money

Next, write each power of 10 in exponential form

At this point, you probably see where we are heading.  The last step is to replace the number 10 by an appropriate variable, say x, and clean up our notation

We now have a true polynomial and can translate characteristics of the polynomial back to characteristics of the number.

A couple of little facts that many people use to test a number for divisibility by 3 or 9 can be derived in this manner.  For those who may not remember those little facts, I will state them here.

A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

In fact, the rules for divisibility by 2 and 5 can also be derived in this manner.

A number is divisible by 2 if and only if the one’s digit is divisible by 2.

A number is divisible by 5 if and only if the one’s digit is 0 or 5.