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### Rational Expressions

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

Hello,

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.

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

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,

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

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

with some specific examples that are giving you trouble.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Polynomials

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