Polynomial Basics and Terms

Date: 05/20/98 at 20:14:28
From: erin
Subject: Algebraic polynomials

I'm having a lot of trouble understanding polynomials. My teacher goes 
through a lesson one chapter a day so it's hard to keep up. If you 
could, I'd like you to help to explain them to me. 

Erin

Date: 05/22/98 at 00:43:35
From: Doctor Santu
Subject: Re: Algebraic polynomials

Well, Erin, polynomials are not much more difficult than "x". As you 
know, in algebra, when we are trying to work backwards to figure out 
an unknown number from how it turns out and what was done to it, we 
call the number x, or whatever, and proceed, right? You use 
polynomials in a similar way.

First, you have to know what x squared, and x cubed and things like 
that are. These are powers of x. Because we have to work with e-mail,
I'm going to write x squared, which is just x times x, as x^2 (though
when I write on paper I write the 2 small and raised a little bit).
Similarly x^3 stands for x*x*x, and so on.

Well. polynomials are just combinations of powers of x, like:

   7x^5 + x^4 - 4x^3 + 15x^2 + 11

That's basically all there is to what they are. If you want more, 
here's some vocabulary that goes with polynomials. (If you haven't 
read or heard any of this, just don't worry. As the great 
mathematician Tallulah Bankhead said, there's less to this than
meets the eye.)

The highest power of x in the polynomial is called the degree of the 
polynomial. The polynomial I wrote above has degree 5. I wrote the 
polynomial in descending order of powers. Not everyone does that; so 
in your homework, look carefully to see if there's a high power of x 
in the middle of the polynomial somewhere.

Each separate part of the polynomial that is combined using + and - 
signs to make up the whole thing is called a term. The above 
polynomial has 5 terms, starting with 7x^5, sometimes called the 
leading term because it's in front, then x^4, then -4x^3, and so on.  
(Minus signs are considered to belong to the term immediately 
following them.)

There are special names for low-powered terms. The term without any x 
at all, in the above case the 11, is called the constant term. The 
term with only x is called the linear term (and our example just 
didn't have one). The term with x^2 is called the quadratic term, 
(15x^2 for our example). The x^3 term is called the cubic term
(-4x^3 in our example), and the fourth-degree term is called the 
quartic term (x^4 in our example).

You can add polynomials, subtract them, and multiply them. Want to 
see?

Here's addition:

   [3x^4 - 4x^3 - 4x^2 + 7x] + [7x^5 + x^4 - 4x^3 + 15x^2 + 11]

   = 7x^5 + 4x^4 - 8x^3 + 11x^2 + 7x + 11

What's the secret?  The terms in one polynomial basically just ignore 
all the terms in the other polynomial except for the one with the 
exact same degree. So the 7x^5, not having any fifth-degree terms in 
the other polynomial, basically remains the same. The 3x^4 in the 
first polynomial and the x^4 in the second, get added together to 
become 4x^4. Think of it as 3 bananas and one banana make 4 bananas.  
The x^4 behaves as if it were an  entirely new unknown as far as 
addition is concerned. Similarly:

   -8x^3 comes from adding -4x^3 and -4x^3,

   11x^2 comes from adding -4x^2 and 15x^2,

   7x is just the 7x from the polynomial on the left, since the 
      polynomial on the right didn't have a term to go with it, and

   11 is from the polynomial on the right because the other polynomial 
      didn't have a constant term.

Multiplication:

   (x^2 + 2x + 3) * (4x^2 + 5x + 6)

You have to multiply each term from the polynomial on the left by
every term on the polynomial on the right, so you get 9 terms in this
example. Then you combine them all together to get the answer:

   4x^4 + 13x^3 + 28x^2 + 27x + 18

Polynomials can be divided by each other. You can do long division of 
one polynomial by another, provided the divisor has a degree that's 
either lower than the one being divided, or maybe equal.

Regards,

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