Polynomials: Terms, Exponents, Degrees

Date: 04/18/2002 at 13:37:28
From: Jordan Ransom
Subject: Math help like cubic stuff.

Can you give me an example of 
 1. Linear binomial
 2. Quadratic trinomial
 3. Cubic trinomial
 4. Quadratic monomial
 5. Six-degree binomial
 6. Quadratic binomial
 7. Linear monomial
 8. Cubic binomial
 9. Fourth-degree trinomial
10. Cubic polynomial with four terms.

Thanks for the help.

Date: 04/18/2002 at 15:07:23
From: Doctor Ian
Subject: Re: Math help like cubic stuff.

Hi Jordan,

First, let's do a little translation.  The following are synonyms:

  linear      <=>  first degree
  quadratic   <=>  second degree
  cubic       <=>  third degree

So now the list looks like this:

   1. First-degree binomial
   2. Second-degree trinomial
   3. Third-degree trinomial
   4. Second-degree monomial
   5. Sixth-degree binomial
   6. Second-degree binomial
   7. First-degree monomial
   8. Third-degree binomial
   9. Fourth-degree trinomial
  10. Third-degree polynomial with four terms.
  
Now let's do another translation.  The following are synonyms:

   monomial     <=>    polynomial with 1 term
   binomial     <=>    polynomial with 2 terms
   trinomial    <=>    polynomial with 3 terms

So now the list looks like this:

   1. First-degree  polynomial  with 2 terms
   2. Second-degree polynomial  with 3 terms
   3. Third-degree  polynomial  with 3 terms
   4. Second-degree polynomial  with 1 term
   5. Sixth-degree  polynomial  with 2 terms
   6. Second-degree polynomial  with 2 terms
   7. First-degree  polynomial  with 1 term
   8. Third-degree  polynomial  with 2 terms
   9. Fourth-degree polynomial  with 3 terms
  10. Third-degree  polynomial  with 4 terms

Now it's starting to look kind of like a table.  

                    Number of terms
  Degree
                  1         2          3          4   

  First          [7]       [1]

  Second         [4]       [6]        [2]

  Third                    [8]        [3]        [10]

  Fourth                              [9]

  Fifth

  Sixth                    [5]

So basically, once you learn how to fill in _any_ slot in the table, 
you know how to fill _all_ of them in.  So even though this looks like 
a bunch of different kinds of things, they're really all just the same 
thing, with a couple of knobs (degree and number of terms) that you 
can tweak.

The 'degree' of a term is the sum of the exponents in the term.  
Here are some examples.

   Term              Exponent(s)     Degree
   ----------        -----------     ------
   2                 0                0          [x^0 = 1]
   2x                1                1
   3x^2              2                2
   3x^2y             2,1              3
   3xy^2             1,2              3
   x^9y^4z^4         9,4,4           17

A polynomial is the sum of a bunch of monomials.  (Note that 'poly' 
means many: a 'polygon' is a shape with many sides, a 'polyglot' is 
someone who speaks many languages, a 'polytheist' believes in many 
gods. And 'mono' means 'one': a 'monogamous' person has only one mate, 
a 'monopoly' is when a product is available from only one vendor, a 
'monotheist' believes in only one god.)

So here are some polynomials:

  2x^2 + 3y

  x^2y + xy^2 + xz - 2x + 4

  x^3

The 'degree' of a polynomial is the _highest_ degree of the monomials 
that make it up. 

What can we do with all this? Well, suppose we want to make a fourth 
degree polynomial with three terms. (This is number 9 on the list.)  
First, we make spaces for the number of terms we want:

  ___ + ___ + ___

Then, we make a monomial with the degree of the polynomial:

    x^4 + ___ + ___

Now just keep adding monomials, being careful not to use an exponent 
higher than the degree of the polynomial:

    x^4 + x + 1

So if this has all made sense to you, you can start cranking out the 
items on your list.  

I hope this helps.  Write back if you'd like to talk more about 
this, or anything else. 

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