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Question

If fraction numerator 2 x plus 1 over denominator left parenthesis x minus 1 right parenthesis open parentheses x to the power of 2 end exponent plus 2 close parentheses end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B x plus c over denominator x to the power of 2 end exponent plus 2 end fraction then B=

  1. 2    
  2. 1    
  3. -1    
  4. -2    

Hint:

Taking LCM on RHS and simplifying , then take any value of x and find a,
Equate the coefficients and find the required value.

The correct answer is: -1


     Given :
    fraction numerator 2 x plus 1 over denominator left parenthesis x minus 1 right parenthesis open parentheses x to the power of 2 end exponent plus 2 close parentheses end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B x plus c over denominator x to the power of 2 end exponent plus 2 end fraction

    Take LCM on RHS
    rightwards double arrow fraction numerator 2 x plus 1 over denominator left parenthesis x minus 1 right parenthesis open parentheses x squared plus 2 close parentheses end fraction equals fraction numerator A open parentheses x squared plus 2 close parentheses space plus left parenthesis B x space plus c right parenthesis space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis open parentheses x squared plus 2 close parentheses end fraction

C a n c e l l i n g space d e n o m i n a t o r s space o n space b o t h space s i d e
space
rightwards double arrow 2 x plus 1 space equals space A open parentheses x squared plus 2 close parentheses space plus left parenthesis B x space plus c right parenthesis space left parenthesis x minus 1 right parenthesis

W h e n space x space equals space 1
rightwards double arrow 2 plus 1 space equals space A open parentheses 1 plus 2 close parentheses space plus space 0
rightwards double arrow space A space equals space 1

E q u a t i n g space c o e f f i c i e n t s space o f space x squared space a n d space x space b y space c o m p a r i n g space L H S thin space a n d space R H S comma space w e space g e t
rightwards double arrow A space plus space B space equals space 0 space rightwards double arrow space B space equals space minus A
rightwards double arrow T h u s comma space B space equals space minus 1

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