Simplifying Algebraic Expressions

Date: 03/23/2002 at 19:01:45
From: Heidi  Callahan
Subject: Simplifying Algebraic Expressions

I am looking for guidelines to follow when simplifying algebraic 
expressions (for simple expressions and more complex expressions).  
Are there certain rules to follow?

Date: 03/27/2002 at 09:12:51
From: Doctor Ian
Subject: Re: Simplifying Algebraic Expressions

Hi Heidi,

In some sense, your algebra textbook is a collection of guidelines for 
simplifying expressions. There's not a lot that I'd be able to add to 
what's already there.

In my experience, the most useful 'rules' are (1) the distributive 
property, and (2) factoring into binomials. 

The former allows you to gather together like terms that are 
originally separated in an expression. The latter allows you to cancel 
identical binomials. For example, suppose we start with a mess like 
this:

   x(xy - 4x) - x(y + 4) - 3(2y + 8)
   --------------------------------
       xy^2 - 16x - 3y^2 + 48

Applying the distributive property allows us to break everything into 
monomials. Once we've done that, we can move like monomials together, 
and use the distributive property again to factor out shared terms.  
The resulting polynomials are candidates for factoring into 
polynomials:


   (x^2y - 4x2) - (xy + 4x) - (6y + 24)   Distribute multiplications
   ------------------------------------    
       xy^2 - 16x - 3y^2 + 48


   x^2y - 4x2 - xy - 4x - 6y - 24         Eliminate parentheses
   ------------------------------          
       xy^2 - 16x - 3y^2 + 48


   x^2y - xy - 6y - 4x2 - 4x - 24         Move like terms together
   ------------------------------         
       xy^2 - 16x - 3y^2 + 48


   y(x^2 - x - 6) - 4(x2 - x - 6)         Factor
   ------------------------------          
       xy^2 - 16x - 3y^2 + 48


      y(x+2)(x-3) - 4(x+2)(x-3)           Factor
   ------------------------------         
       xy^2 - 16x - 3y^2 + 48


          (y-4)(x+2)(x-3)                 Undistribute multiplications
   ------------------------------        
       xy^2 - 16x - 3y^2 + 48


          (y-4)(x+2)(x-3) 
   ------------------------------          
       x(y^2 - 16) - 3(y^2 - 16)          Factor


          (y-4)(x+2)(x-3) 
   ------------------------------    
          (x-3)(y^2 - 16)                 Undistribute multiplications


          (y-4)(x+2)(x-3) 
   ------------------------------          
          (x-3)(y+4)(y-4)                 Factor



              (x+2) 
   ------------------------------         Cancel
              (y+4)
   


All of which is to say, armed with only these two techniques, you can 
go a long, long way. Which would explain why so much emphasis is 
placed on them!

I hope this helps.  Write back if you'd like to talk more about this, 
or anything else.

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