Monomials, Polynomials

Date: 03/02/2001 at 13:31:35
From: Chris Heller
Subject: Monomials

Can you explain monomials and polynomials to me?

Date: 03/02/2001 at 17:57:10
From: Doctor Jordi
Subject: Re: Monomials

Hello, Chris -  thanks for writing to Dr. Math.

I am not sure exactly what you would like to know about monomials and 
polynomials. There is not much to say about monomials, except that 
they are the building blocks of polynomials. Polynomials, on the other 
hand, are very interesting mathematical structures, and a whole lot of 
things can be said about them. I will spare you for the moment saying 
every interesting fact about polynomials that I can think of, since I 
am imagining that for now you are only interested in knowing the 
definitions regarding the two.

A monomial is a product of as many factors as you like, each raised to 
a POSITIVE power.  By definition, negative exponents are not allowed.
The following are examples of monomials:

  x
  y
  x^2                (x squared)
  3xy
  34.6*(a^3)*r^4
  3(x^26)*(y^(pi))  (3 times x to the 26th power times y to the pi power)
  37*a*b*c*d* ... *z*alpha*beta*gamma* ... *omega*aleph*(a partridge in
                                                        a pear tree)

(The product of 37 times the variables represented by all the letters 
of the English alphabet times the variables represented by all the 
letters of the Greek alphabet times the first letter of the Hebrew 
alphabet, times a partridge in a pear tree. It is still a monomial, no 
matter how many numbers we are multiplying together. Don't worry too 
much about this goofy example; I am just trying to point out that a 
monomial is a very general concept.)

A polynomial is nothing more than the sum of two or more monomials.  
The following are examples of polynomials:

  x                    (yes, a monomial is also a polynomial)
  x + y
  ax^2 + bx + c        (the very famous quadratic polynomial)
  (x^2)*(y^2) + xy + x
  x^34 + r^(3.535) + c
  abc + rst + xyz^2 + abcdefghijklmnopqrstuvwxyz

Again, don't worry too much about that last goofy example.  I'm sure 
you will probably never encounter a polynomial anytime soon that uses 
all the letters of the English alphabet.

Usually, however, when we say "polynomial," we are talking about a 
very specific kind of polynomial that occurs very often. This is the 
polynomial where there is only one variable, all the powers of this 
variable are nonnegative integers, and each power of this variable may 
be multiplied by a constant. The way mathematicians usually write this 
polynomial is

 A_n * x^n + A_(n-1) * x^(n-1) + ... + A_2 * x^2 + A_1 * x + A_0

Where the A_i symbols (A_i is intended to represent A with a subscript 
i, the letter A indexed by i) represent constants (i.e. numbers like 
0, 1, -3, 7/4, 34.334 and pi) and n is a positive integer. The famous 
quadratic polynomial I mentioned above in my examples is a special 
case of these very common polynomials when n = 2.  

In these polynomials, by the way, n is called the "degree" of the 
polynomial. Thus, a quadratic polynomial has degree 2. Polynomials of 
degree 3 are called "cubic polynomials," of degree 1 are called 
"linear expressions" (they are technically also polynomials, but 
nobody calls them that) and of degree 0 (i.e., there are no variables, 
only the A_0) are called "constants."  Yes, constants are also a 
special case of polynomials. In fact, a constant by itself is also a 
monomial, but we usually never call them that, in order to avoid 
confusion.  

Summarizing, a monomial is a single mathematical expression that 
contains only multiplication and exponentiation. A polynomial is a sum 
of monomials, so you can think of it as a mathematical expression that 
only contains the operations of addition, multiplication, and 
exponentiation.

I hope this explanation helped.  Please write back if you have more 
questions, or if you would like to talk about this some more.

- Doctor Jordi, The Math Forum
  http://mathforum.org/dr.math/