Algebra: Factoring Fractions

Date: 07/30/97 at 02:13:05
From: Wendy Herndon
Subject: Algebra 2

  8ab                16b2
------  divided by  ---------
4a2-b2             8a2 - 4ab   
  
My sister showed me how to do this but I couldn't understand. I think 
where I'm stuck is where it came to factoring. 

If you could help me I'd be so happy.

Thanks, Wendy

Date: 07/30/97 at 07:43:38
From: Doctor Anthony
Subject: Re: Algebra 2

When dividing by a fraction, you can write it more clearly by 
MULTIPLYING by that fraction turned upside down. So the whole 
expression is:

       8ab           8a^2 - 4ab
    -----------  x  ------------
   4a^2 - b^2          16b^2


The term  (4a^2-b^2)  is a difference of squares and factorizes to
(2a-b)(2a+b).  You can prove this to your own satisfaction by 
multiplying out the two brackets:

(2a-b)(2a+b) =  4a^2 + 2ab - 2ab - b^2  =  4a^2-b^2

       8ab x 4a(2a - b)            32a^2.b(2a-b)
      ------------------    =    ------------------
     (2a-b)(2a+b) x 16b^2        16b^2(2a-b)(2a+b)

Now you can cancel any terms that are common to the top and bottom 
lines. The bracket (2a-b) occurs on both lines and can be cancelled.  
The number 32 on the top line is reduced to 2 by the 16 on the bottom 
line. Also there is a factor b on both lines, and this can be 
cancelled. After these cancellations we get:

          2a^2
        ---------
         b(2a+b) 

This is as far as the simplification can be taken.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/