Expert: Paul Klarreich - 3/11/2011

Question
let f(x)=(1-x)²/(1+x)(1+x²),express f(x) in partial fraction.  hence or otherwise,show taat if x is sufficiently small for power of x above the second to be neglected,then f(x)=1-3x+3x².

Answer
Questioner: rusooly  
Country: Mauritius
Category: Advanced Math
Private: Yes  <<<<<<<<<<<<<<<<<<<<<<< changed.
Subject: partial fractions and binomial expansions
Question: let f(x)=(1-x)²/(1+x)(1+x²),express f(x) in partial fraction.  hence or otherwise,show taat if x is

sufficiently small for power of x above the second to be neglected,then f(x)=1-3x+3x².
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Set up the partial fractions:

 1 - 2x + x^2       A        Bx + C
---------------- = ------ + -----------
(1 + x)(1 + x^2)   1 + x     1 + x^2

Multiply out:
 1 - 2x + x^2  =   A(1 + x^2) +  (Bx + C)(1 + x)

Now do the usual stuff:

A + C = 1
B + C = -2
A + B = 1

C = -1

A = 2

B = -1

 1 - 2x + x^2       2        x + 1
---------------- = ------ - -----------
(1 + x)(1 + x^2)   1 + x     1 + x^2

Now you want to assume  x^2 is 'negligible', as those physicists say:

 1 - 2x + x^2       2       x + 1
---------------- = ------ - -------
(1 + x)(1 + x^2)   1 + x       1  


   2  - (x + 1)^2
= -----------------
      1 + x


   1  - x^2 - 2x
= ----------------
      1 + x

Now you want to eliminate the '+ x':

Rationalize:

   1  - x^2 - 2x  1 - x
= ---------------- ------
      1 + x       1 - x

   1  + x^2 - 3x + x^3
= -----------------------
          1 - x^2

Now if you discard the x^2 terms, you get your answer.(I have x^3 instead of x^2)