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Restrictions

Date: 04/16/2003 at 20:55:32
From: Eric
Subject: Factoring and Restrictions

Consider this expression

x(z - 5)        a
--------  *  --------
   y         b(z - 5)

We can say that it is equal to:

   xa
   --
   yb

but z cannot equal 5.

But what is the point of restrictions if I can create any one that I 
want?

I could say that:

     xa
     --
     yb


     x(z - 5)(z - 4)(z - 3)               a
 =   ----------------------  *  ----------------------
               y                b(z - 5)(z - 4)(z - 3)

and the restrictions are that x cannot equal 3, 4, or 5, so there is 
really no point in them.

Can you explain this?


Date: 04/17/2003 at 09:07:58
From: Doctor Peterson
Subject: Re: Factoring and Restrictions

Hi, Eric.

You say in your subject line that you are talking about factoring, 
but that is not what you are doing; you are simplifying expressions.

When you simplify a rational expression by canceling terms, you are 
REMOVING an implicit restriction, so you can't properly say that the 
new expression is fully equivalent to the original; that's why you 
have to add an explicit restriction to the new expression: the new 
one is equal to the old one WHEN z is not 5, otherwise the old one is 
undefined but the new one is. So stating the restriction explicitly 
only keeps you aware of a restriction that was present in the 
original expression, but is no longer in the new expression.

If you introduce new (cancelable) factors into an expression, you are 
adding new implicit restrictions that were not there to begin with, 
so you are not making an equivalent expression at all. It would only 
be equivalent if the original expression had been given WITH explicit 
restrictions. Do you see the difference?

The point is that if you claim that two expressions are equivalent, 
they must have the same domain (set of values for which they are 
defined); that domain can be specified implicitly (by the fact that 
you can't divide by zero, for instance) or explicitly (by stating 
restrictions), but they must match.

Does that help?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 04/21/2003 at 00:29:30
From: Eric
Subject: Thank you (Factoring and Restrictions)

Thank you Doctor Peterson.  This makes sense to me now.  I see now 
that we are recording the restrictions to keep track of the 
difference between the old and new expressions.
Associated Topics:
High School Polynomials

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