Monomials, Polynomials

Date: 01/21/98 at 19:07:46
From: Danick Baron
Subject: Algebra

Dear Dr. Math,

I'm a student in High School who really needs your help. I'm having 
trouble with the following:

1. multiplying monomials:
   A) (nm) (nm)   B) (-3ab)to the third power(2b)

2. dividing monomials:
   A)12d/4d(to the 5th power)  B)  24x(2power)y(3power)z / -6xyz

3. adding monomials:
   A) (7A)+(3A)               

4. subtracting monomials:
   A) (-9A)-(3A)

5. adding and subtracting polynomials
   A) (5ax+3ax-7a)+(2ax+8ax+4)   B) (6xy-3xy-7)-(5xy+2xy+3)

Danick2@Juno.Com

Date: 01/26/98 at 10:10:20
From: Doctor Sonya
Subject: Re: Algebra

Hi, Danick.

Your problems all have to do with monomials and polynomials. The first 
thing you have to know is what these are. A polynomial is an 
expression like:

   x^2 + x - 5  (x^2 means x squared).

In fact, all polynomials are sums of terms like a(x^b) where a and b 
are numbers and x is some unknown variable. Another example of a 
polynomial is:

    x^5 - x^3 + 44x - 32

Here is an example of where I could use a polynomial in real life.  
Let's say I have a factory that produces 100 stuffed elephants a day 
(not quite as boring as a cardboard box factory). But let's also say 
that for Christmas, I decided to give 4 away to charity. Now I want 
to know how many elephants I produced since the beginning of December.  
I can write down a polynomial that will describe the number for me, 
where I let x be the number of working days since December first:

   100x - 4

I have the 100x, because every day I produce 100, and the -4 counts 
the four that I gave away at Christmas. The power of this polynomial 
is that I can use the same polynomial on any day (for any x) and find 
out how many elephants I've produced. This unknown is what makes them 
so important.

A monomial is just like a polynomial, but it only has one term in it.  
Some examples of monomials are:

   3x
   34x^5
   2ab (here my variables are a and b, not x)

To manipulate polynomials and monomials, there are some simple rules 
to follow. The first is to combine like terms. What this means is to 
take all the terms with the same variable in them and combine them.  
When looking for the same variables, remember that the x is not the 
same as x^2.  One way to think about it is to replace the variable 
with the word "whale" (you can actually choose any word you want. I'm 
just on a large mammal kick today).  Then the polynomial:

   4A + 6A

becomes:

   4 whales plus 6 whales.

Well, you know how to add, and this equals:

   10 whales.

Now, change the whales back to A's, and you see that 
   4A + 6A = 10A. 

For another example, if I had:

   x^2 + 3 - x + 4 + 2x^2

I would group together all of the x^2 and all of the numbers with no 
variables (called constants):

   (x^2 + 2x^2) - x + (3 + 4)

Now, combining the like terms, I get

   3x^2 - x + 7

You can use this rule to find the answers to questions 3, 4, and 5.

Multiplying monomials is even easier.  Just treat the variables as if 
they were numbers.  For example, if I had to multiply:

   3*4*4*2*7*2   (* means multiply)

I could group the numbers that are the same together and rewrite it 
as:

   2^2 * 3 * 4^2 * 7

If I had a variable thrown in, I would just treat it like the numbers:

   3*4*4*X*2*2*7
 = 2^2 * 3 * X * 4^2 * 7

Here's another example:

  (3*a*b)(4)(32*a)(b)

Now, I combine like terms:

  (3*4*32)(a*a)(b*b)
 =  384 * a^2 * b^2 

This should help you with question 1. 

Remember that if you have a monomial and you raise it to some power, 
you are just multiplying it by itself that number of times. For 
example, 

   (2xy)^3 = (2xy)(2xy)(2xy)

Now, for dividing monomials. This is actually my favorite. What you do 
is set it up as a fraction, and then cancel anything that is the same 
on the top and the bottom.  For example, if I had:

   4 * a * b
  -----------
   2 * a^2 

I could rewrite it as:

  2 * 2 * a * b
  -------------
    2 * a * a

and then begin to cancel. I would cancel one two from the top and one 
from the bottom, and get:

  2 * a * b
  ---------
    a * a

Then I would cancel an a from the top and the bottom:

  2 * b
  -----
    a

I can't cancel anything else, and the monomial is simplified.

I'll go through one of your examples, and perhaps this will seem 
clearer.

  12d/4d(to the 5th power)

I can rewrite this as:

   12 * d
   ------
   4 * d^5

And rewrite this as:

        4 * 3 * d
  -----------------------
  4 * d * d * d * d * d 

Now, I can cancel 4 from the top and bottom, and and d from the top 
and bottom.  That gives me:

     3 * d
  -------------
  d * d * d * d

Which equals:

   (3d)/d^4.

One thing to watch out for is that you can only cancel things if they 
are MULTIPLIED by everything else in that part of the fraction.  For 
example, if I have:

  4y + x
  -------
    x^2

I can't cancel an x, because the x in the numerator is not multiplied 
by everything else, it's added.

If for whatever reason you still need more help, please write us back.

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