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### Polynomial Basics and Terms

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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
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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?

[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

-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/
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Associated Topics:
High School Definitions
High School Polynomials

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