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Rational Expressions

Date: 01/06/2002 at 20:33:56
From: Ashley
Subject: Rational Expressions


Basically I just need some help on rational expressions in general.
My main problems are:

1) How do I know when I need to factor and when I don't, and if there 
are the same polynomials, trinomials, etc., can I cancel them out at 
all times?

2) What are the restrictions on simplified expressions? I always seem 
to do something wrong.

Thanks for your help.
Ashley Townsend

Date: 01/07/2002 at 09:24:35
From: Doctor Ian
Subject: Re: Rational Expressions

Hi Ashley,

The answer to your first question is: You want to factor whenever you 
can. In the best case, it helps you identify terms that cancel each 
other out; in the worst case, you end up with an expression whose 
meaning is more plainly visible. 

The answer to your second question is: You can never, ever, under any 
circumstances, divide by zero. What makes this tricky is that you 
might start with an expression like:

  x^2 - 7x + 12
       x - 3

and simplify it like this:

  x^2 - 7x + 12   (x - 3)(x - 4)
  ------------- = -------------- = x - 4
       x - 3          (x - 3)

Now, what does this _mean_? It means that at any value of x EXCEPT 3, 
you can use the simple expression (x - 4) in place of the more 
complicated expression (x^2 - 7x + 12)/(x - 3).  

But the original expression is undefined at x = 3, so the simpler 
expression - when used in place of the complicated expression - is 
also undefined there. 

(Think of it this way. Suppose I lend you my car, and I say: You can 
do whatever you want, as long as you don't drive the car in New 
Jersey. Now, suppose you lend the car to a friend. You have to tell 
him that HE can't drive it in New Jersey, either! And if he lends the 
car to another friend, he has to pass the warning along.) 

To keep track of this kind of thing, whenever you cancel out something 
like (x - 3) from a denominator, you can make a note off to the side, 
to help you remember the exceptions later on:

  x^2 - 7x + 12   (x - 3)(x - 4)
  ------------- = -------------- 
       x - 3          (x - 3)

                = (x - 4)         [undefined at x = 3]

If I haven't addressed your question completely, please write back 
with some specific examples that are giving you trouble. 

Does this help? 

- Doctor Ian, The Math Forum   
Associated Topics:
High School Polynomials

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