Adding Algebraic Fractions, Different Denominators

Date: 8/24/96 at 17:41:49
From: Anonymous
Subject: Adding Fractions with Different Denominators

My problem is:

7   2-x    3+5x
- + ---  + ----
2x  3x^2   6x^3

How on earth do I solve this? Help!

Date: 8/26/96 at 12:30:58
From: Doctor Mike
Subject: Re: Adding Fractions with Different Denominators

Hello Louie,
  
It isn't really that hard.  It's just that the original problem
has you adding apples and oranges and mangoes, so to speak.  I am
going to show you how to convert this to a problem where you are
adding all mangoes, and then the result will be mangoes, what else!
  
The secret is to get a Common Denominator.  If you have fractions
to add and all the denominators are EXACTLY the same, then you just
use that denominator which is common to all the fractions as the
denominator for your result, and add the numerators to get the 
numerator of the result.  In symbols, a/x + b/x + c/x = (a+b+c)/x .
  
The common denominator in your problem is 6x^3.  The way you get the
other 2 fractions to have the same denominator is to multiply each of
them by one - not just any old one, but a clever version of one:
  
       7   3x^2    2-x   2x    3+5x     21x^2    4x-2x^2   3+5x 
       --*------ + ----*---- + ----  =  ------ + ------- + ----  
       2x  3x^2    3x^2  2x    6x^3      6x^3     6x^3     6x^3 
  
Now you can add and simplify the numerators to get the answer.  See?
I'm sure you can finish it off.  I hope this helps. 
  
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/