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Question

If fraction numerator x to the power of 2 end exponent plus x plus 1 over denominator x to the power of 2 end exponent plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis to the power of 2 end exponent end fraction then A-B=

  1. 4C    
  2. 2    
  3. 3C    
  4. 4C+1    

Hint:

Take LCM on left hand side and simplify. Then equate the coefficients of RHS and LHS

The correct answer is: 2


     Given :
     fraction numerator x to the power of 2 end exponent plus x plus 1 over denominator x to the power of 2 end exponent plus 2 x plus 1 end fraction equals A plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator left parenthesis x plus 1 right parenthesis to the power of 2 end exponent end fraction
    Taking LCM on left hand side and simplifying
    rightwards double arrow fraction numerator x squared plus x plus 1 over denominator x squared plus 2 x plus 1 end fraction equals fraction numerator A left parenthesis x plus 1 right parenthesis squared space plus space B left parenthesis x plus 1 right parenthesis space plus space C over denominator left parenthesis x plus 1 right parenthesis squared end fraction

rightwards double arrow fraction numerator x squared plus x plus 1 over denominator left parenthesis x plus 1 right parenthesis squared end fraction equals fraction numerator A left parenthesis x squared plus 2 x space plus space 1 right parenthesis space plus space B left parenthesis x plus 1 right parenthesis space plus space C over denominator left parenthesis x plus 1 right parenthesis squared end fraction

rightwards double arrow fraction numerator x squared plus x plus 1 over denominator blank end fraction equals fraction numerator A left parenthesis x squared plus 2 x space plus space 1 right parenthesis space plus space B left parenthesis x plus 1 right parenthesis space plus space C over denominator blank end fraction
    rightwards double arrow space x squared space plus space x space plus space 1 space equals space A x squared space space plus space x left parenthesis 2 A space plus space B right parenthesis space plus space A space plus space B space plus space C
C o m p a r i n g space a n d space E q u a t i n g space t h e space c o e f f i c i e n t s space o f space L H S thin space a n d space R H S

A space equals space 1 space a n d space 2 A space plus space B space equals space 1 space
A equals space 1 space a n d space B space equals space minus 1
A space plus space B space plus space C space equals space 1 space rightwards double arrow C space equals space 1
    A space minus space B space equals space 1 minus space left parenthesis negative 1 right parenthesis space equals space 2

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