Quandaries
and Queries 

I am a teacher for grades 1012. Name: Bruce Questions:


Hi Bruce, There are some web pages maintained by Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida, that are a wonderful resource for historical questions and questions about mathematical termonology. Go to his page, Earliest Known Uses of Some of the Words of Mathematics and search under P for polynomial. Polynomials, defined as you have done above, have many nice properties. One of these is that the domain of a polynomial is the entire real line. That is if f(x) is a polynomial and s is a real number then f(s) is a real number. On the other hand if f(x) has a term with a negative power of x then f(0) is not a real number.For example if f(x) = x^{1} then f(0) is undefined and the graph of f(x) is in two pieces. f(x) = x^{1} Polynomials play a role in the development of functions, somewhat analogous to the role that the integers play on the real line.If you add, subtract or multiply two polynomials you get another polynomial. We know that, with the integers, if we want to allow division of any two integers then we must expand our numbers to include the rationals. Even here there is a problem since division by zero does not result in a number. Likewise with polynomials, if we want to allow division of any two polynomials then we must expand the class of functions and, as in the example above of the polynomial 1 divided by the polynomial x, the resulting function may produce outputs that are not numbers.
The functions of the form f(x) = a_{n}x^{n} +
a_{n1}x^{n1} + ... + a_{1}x + a_{0} Harley 

