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Question

$fraction numerator 3 x to the power of 2 end exponent plus x plus 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end exponent end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B over denominator left parenthesis x minus 1 right parenthesis to the power of 2 end exponent end fraction plus fraction numerator C over denominator left parenthesis x minus 1 right parenthesis to the power of 3 end exponent end fraction plus fraction numerator D over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end exponent end fraction$ then $A plus B minus C plus D equals$

0
15
1
10

hint Hint:

Take common LCM on RHS and simplify.

The correct answer is: 1

Given :
$fraction numerator 3 x squared plus x plus 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction equals fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B over denominator left parenthesis x minus 1 right parenthesis squared end fraction plus fraction numerator C over denominator left parenthesis x minus 1 right parenthesis cubed end fraction plus fraction numerator D over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction$
Taking LCM on RHS
$rightwards double arrow fraction numerator 3 x squared plus x plus 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction equals fraction numerator A left parenthesis x minus 1 right parenthesis cubed space plus space B left parenthesis x minus 1 right parenthesis squared space plus space C left parenthesis x minus 1 right parenthesis space plus space D over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction I f space w e space s o l v e space t h i s space f u r t h e r space w e space w i l l space g e t space c o m p l i c a t e d space e q u a t i o n space i n s t e a d rightwards double arrow L e t space left parenthesis x minus 1 right parenthesis space equals space y space rightwards double arrow space x space equals space y space plus space 1 rightwards double arrow L H S space equals space fraction numerator 3 x squared plus x plus 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction equals fraction numerator 3 left parenthesis y plus 1 right parenthesis squared plus y plus 1 plus 1 over denominator left parenthesis y plus 1 minus 1 right parenthesis to the power of 4 end fraction space equals space fraction numerator 3 left parenthesis y squared space plus 2 y space plus 1 right parenthesis space plus space y space plus 2 over denominator left parenthesis y right parenthesis to the power of 4 end fraction space equals space fraction numerator 3 y squared space plus space 7 y space plus 5 over denominator y to the power of 4 end fraction rightwards double arrow space fraction numerator 3 y squared space over denominator y to the power of 4 end fraction plus space fraction numerator 7 y space over denominator y to the power of 4 end fraction space plus space fraction numerator 5 space over denominator y to the power of 4 end fraction space equals space fraction numerator 3 over denominator y squared space end fraction plus space fraction numerator 7 space over denominator y cubed end fraction space plus space fraction numerator 5 space over denominator y to the power of 4 end fraction space rightwards double arrow space fraction numerator 3 x squared plus x plus 1 over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction space equals space fraction numerator 3 over denominator left parenthesis x minus 1 right parenthesis squared space end fraction plus space fraction numerator 7 space over denominator left parenthesis x minus 1 right parenthesis cubed end fraction space plus space fraction numerator 5 space over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction space O n space C o m p a r i n g space fraction numerator 0 over denominator x minus 1 end fraction plus fraction numerator 3 over denominator left parenthesis x minus 1 right parenthesis squared space end fraction plus space fraction numerator 7 space over denominator left parenthesis x minus 1 right parenthesis cubed end fraction space plus space fraction numerator 5 space over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction space space equals space fraction numerator A over denominator x minus 1 end fraction plus fraction numerator B over denominator left parenthesis x minus 1 right parenthesis squared end fraction plus fraction numerator C over denominator left parenthesis x minus 1 right parenthesis cubed end fraction plus fraction numerator D over denominator left parenthesis x minus 1 right parenthesis to the power of 4 end fraction A space equals space 0 comma space B space equals space 3 comma space C space equals space 7 space a n d space D space equals space 5 space$

Thus, A + B - C + D = 0 +3 -7 +5 = 1

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