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open vertical bar table row cell a to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell a b end cell cell c a end cell row cell a b end cell cell b to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell b c end cell row cell c a end cell cell b c end cell cell c to the power of 2 end exponent plus x to the power of 2 end exponent end cell end table close vertical bar is divisor of

  1. a to the power of 2 end exponent    
  2. b to the power of 2 end exponent    
  3. c to the power of 2 end exponent    
  4. x to the power of 2 end exponent    

The correct answer is: x to the power of 2 end exponent


    open vertical bar table row cell a to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell a b end cell cell c a end cell row cell a b end cell cell b to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell b c end cell row cell c a end cell cell b c end cell cell c to the power of 2 end exponent plus x to the power of 2 end exponent end cell end table close vertical bar
    Multiply C subscript 1 end subscript comma C subscript 2 end subscript comma C subscript 3 end subscript by a comma b comma crespectively and hence divide by abc
    \ capital delta equals fraction numerator 1 over denominator a b c end fraction open vertical bar table row cell a left parenthesis a to the power of 2 end exponent plus x to the power of 2 end exponent right parenthesis end cell cell a b to the power of 2 end exponent end cell cell c to the power of 2 end exponent a end cell row cell a to the power of 2 end exponent b end cell cell b left parenthesis b to the power of 2 end exponent plus x to the power of 2 end exponent right parenthesis end cell cell b c to the power of 2 end exponent end cell row cell c a to the power of 2 end exponent end cell cell b to the power of 2 end exponent c end cell cell c left parenthesis c to the power of 2 end exponent plus x to the power of 2 end exponent right parenthesis end cell end table close vertical bar
    Now take out a, b and c common from R subscript 1 end subscript comma R subscript 2 end subscript and R subscript 3 end subscript,
    \ capital delta equals open vertical bar table row cell a to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell b to the power of 2 end exponent end cell cell c to the power of 2 end exponent end cell row cell a to the power of 2 end exponent end cell cell b to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell c to the power of 2 end exponent end cell row cell a to the power of 2 end exponent end cell cell b to the power of 2 end exponent end cell cell c to the power of 2 end exponent plus x to the power of 2 end exponent end cell end table close vertical bar
    Now applying C subscript 1 end subscript rightwards arrow C subscript 1 end subscript plus C subscript 2 end subscript plus C subscript 3 end subscript
    rightwards double arrow capital delta equals left parenthesis a to the power of 2 end exponent plus b to the power of 2 end exponent plus c to the power of 2 end exponent plus x to the power of 2 end exponent right parenthesis open vertical bar table row 1 cell b to the power of 2 end exponent end cell cell c to the power of 2 end exponent end cell row 1 cell b to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell c to the power of 2 end exponent end cell row 1 cell b to the power of 2 end exponent end cell cell c to the power of 2 end exponent plus x to the power of 2 end exponent end cell end table close vertical bar
    Þ capital delta equals x to the power of 4 end exponent left parenthesis a to the power of 2 end exponent plus b to the power of 2 end exponent plus c to the power of 2 end exponent plus x to the power of 2 end exponent right parenthesis
    Hence, it is divisible by x to the power of 2 end exponent

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    open vertical bar table row cell b to the power of 2 end exponent plus c to the power of 2 end exponent end cell cell a to the power of 2 end exponent end cell cell a to the power of 2 end exponent end cell row cell b to the power of 2 end exponent end cell cell c to the power of 2 end exponent plus a to the power of 2 end exponent end cell cell b to the power of 2 end exponent end cell row cell c to the power of 2 end exponent end cell cell c to the power of 2 end exponent end cell cell a to the power of 2 end exponent plus b to the power of 2 end exponent end cell end table close vertical bar equals

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