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### 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
3. Cubic trinomial
5. Six-degree 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
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

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/
```
Associated Topics:
High School Polynomials

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