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Asked by araima2001 | 19th Nov, 2016, 07:07: PM

Expert Answer:

begin mathsize 16px style Let space the space roots space of space straight x cubed minus ax squared plus bx minus straight c equals 0 space be space straight alpha comma space minus straight alpha space and space straight gamma.
So comma space let space straight a equals 1 comma space straight b equals negative straight a comma space straight c equals straight b space and space straight d equals negative straight c
We space know space that space for space straight a space cubic space equation comma
Sum space of space the space roots equals negative straight b over straight a equals negative fraction numerator left parenthesis negative straight a right parenthesis over denominator 1 end fraction equals straight a
rightwards double arrow straight alpha plus left parenthesis negative straight alpha right parenthesis space plus space straight gamma space equals space straight a
rightwards double arrow straight gamma space equals space straight a.... left parenthesis straight i right parenthesis
Sum space of space the space product space of space the space roots space taken space two space at space straight a space time equals straight c over straight a space equals straight b over 1 equals straight b
rightwards double arrow αβ plus βγ plus αγ equals straight b
rightwards double arrow straight alpha left parenthesis negative straight alpha right parenthesis plus left parenthesis negative straight alpha right parenthesis straight gamma plus αγ equals straight b
rightwards double arrow negative straight alpha squared equals straight b.... left parenthesis ii right parenthesis
Product space of space the space roots space equals negative space straight d over straight a
rightwards double arrow αβγ equals negative space fraction numerator left parenthesis negative straight c right parenthesis over denominator 1 end fraction
rightwards double arrow straight alpha left parenthesis negative straight alpha right parenthesis straight gamma equals space straight c
rightwards double arrow negative straight alpha squared straight gamma equals straight c.... left parenthesis iii right parenthesis
Substituting space left parenthesis straight i right parenthesis comma space left parenthesis ii right parenthesis space in space left parenthesis iii right parenthesis comma space we space get
minus straight alpha squared straight gamma equals straight c space that space is comma left parenthesis straight b right parenthesis left parenthesis straight a right parenthesis equals straight c
rightwards double arrow ab equals straight c
Hence comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Answered by Rebecca Fernandes | 20th Nov, 2016, 02:00: PM