Definition of a Polynomial

Date: 12/04/97 at 17:06:23
From: Richard
Subject: Definition of a Polynomial

My daughter had a problem that said "are the following expressions 
polynomials?" Most were straightforward but one was just the number 5. 
She said no and I agreed. She got it wrong. I called a professor of 
math at a local university, who explained that it is a polynomial 
because it is 5 times x^0 (x to zero power.)  

Okay, I get that. But one of the other problems was just x. The answer 
in the back of the book says no, x alone is not a polynomial. How is 
just x different from just 5?

Thanks!

Date: 12/04/97 at 18:58:58
From: Doctor Pete
Subject: Re: Definition of a  Polynomial

Hi,

I'd say the book was wrong. For a more detailed answer, most
mathematicians would define a polynomial this way:

A polynomial of one variable is a function P[x] of the form 
a[0] + a[1]x^1 + a[2]x^2 + ..., where the coefficients a[0], a[1], ... 
may be 0. The largest value of n for which a[n] is not zero is called 
the degree of the polynomial. More generalized polynomials occur when 
we replace x with some function of x, say Sin[x] or Exp[x]. There are 
also polynomials of more than one variable, which you can guess have 
individual terms of the form

     a[k] x1^(p1) x2^(p2) ... xm^(pm).

Clearly, the value x falls under this category (a[1] = 1, a[k] = 0
otherwise), and so does 5 (a[0] = 5, a[k] = 0 otherwise). So does

     1+x+x^2+x^3+x^4+...  = 1/(1-x),

which we usually don't like to call a polynomial since it has infinite
degree. Rather, we call it a power series.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 12/05/97 at 01:05:40
From: Richard
Subject: Re: Definition of a  Polynomial

Thanks. Much useful information. Really appreciate the quick response. 
Thanks again!