Dear Durgesh,

Express as a partial fraction


	 x - 1
    ---------------
    (3x - 5)(x - 3)


========================

Solution:

First write the fraction as

	  x - 1               A                B
     ---------------   =   --------    +    -------
     (3x - 5)(x - 3)       (3x - 5)         (x - 3)


Notice that I have taken the two terms that are in brackets and placed them on their own
as seperate fractions using ''A'' and ''B'' as numerators. 

A and B are the mystery numbers we need to discover.

Next, multiply denominator (the bottom bit of the big fraction) by both sides of the equation to cancel out terms



So we now have

	  x - 1       =   A(x - 3)   +   B(3x - 5)


In order to get rid of either the ''A'' or ''B'' term, we can substitute a ''strategic'' value to make it equal to zero. 
For example, to get rid of A(x - 3), let''s make x = 3, so (3 - 3) ends up as 0. Gone.

Remember that you have to do this to all of the x''s in the equation

	  3 - 1	      =   A(3 - 3)  +  B(3*3 - 5) you can see the ''A'' component becomes zero.

	  3 - 1	      =   B(9 - 5)

	  2	      =   4B

   so,	 2/4          =   B

   or,	 1/2          =   B


We''ve discovered what B is, now we can repeat the process to discover A

Let''s make x = 5/3 to get rid of the ''B'' part of the equation...

so,          B(5/3 * 3) = 5

and,         B(5 - 5)   = 0

The B(3x - 5) term is gone! So we can plug this value into the entire equation to find A


	  5/3 - 1      =   A(5/3 - 3)

	  2/3          =   A * -4/3

	  2/3
         ------        =   A
          -4/3

	  -1/2         =   A



We now have A and B, and the answer to our problem.


	 x - 1               -1              1
    ---------------   =   ---------   +   --------
    (3x - 5)(x - 3)       2(3x - 5)       2(x - 3)

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