What is the correct option from the given question? ps - I got 98 as an answer but it's not correct(? idk maybe the given answer at the back is wrong, u can also see and recheck my solution, I've attached the solution below the question)

2 Answers
George C.
Dec 20, 2017

#98# is the correct answer.

Explanation:

Given:

#4x^3-7x^2+1 = 0#

Dividing by #4# we find:

#x^3-7/4x^2+0x+1/4#

#= (x-alpha)(x-beta)(x-gamma)#

#= x^3-(alpha+beta+gamma)x^2+(alphabeta+betagamma+gammaalpha)x-alphabetagamma#

So:

#{ (alpha+beta+gamma = 7/4), (alphabeta+betagamma+gammaalpha = 0), (alphabetagamma = -1/4) :}#

So:

#49/16 = (7/4)^2-2(0)#

#color(white)(49/16) = (alpha+beta+gamma)^2-2(alphabeta+betagamma+gammaalpha)#

#color(white)(49/16) = alpha^2+beta^2+gamma^2#

and:

#7/8 = 0 - 2(-1/4)(7/4)#

#color(white)(7/8) = (alphabeta+betagamma+gammaalpha)^2-2alphabetagamma(alpha+beta+gamma)#

#color(white)(7/8) = alpha^2beta^2+beta^2gamma^2+gamma^2alpha^2#

So:

#49/128 = (7/8)^2-2(-1/4)^2(49/16)#

#color(white)(49/128) = (alpha^2beta^2+beta^2gamma^2+gamma^2alpha^2)^2-2(alphabetagamma)^2(alpha^2+beta^2+gamma^2)#

#color(white)(49/128) = alpha^4beta^4+beta^4gamma^4+gamma^4alpha^4#

So:

#alpha^(-4)+beta^(-4)+gamma^(-4) = (alpha^4beta^4+beta^4gamma^4+gamma^4alpha^4)/(alphabetagamma)^4#

#color(white)(alpha^(-4)+beta^(-4)+gamma^(-4)) = (49/128)/(-1/4)^4#

#color(white)(alpha^(-4)+beta^(-4)+gamma^(-4)) = (49/128)/(1/256)#

#color(white)(alpha^(-4)+beta^(-4)+gamma^(-4)) = 98#

George C.
Dec 20, 2017

#98#

Explanation:

Alternatively, as an extra check, note that the roots of:

#4x^3-7x^2+1 = 0#

are the reciprocals of the roots of:

#x^3-7x+4 = 0#

So we can find #alpha^4+beta^4+gamma^4# for the roots of this cubic in order to calculate #alpha^(-4)+beta^(-4)+gamma^(-4)# for the roots of the original cubic.

Given:

#x^3+0x^2-7x+4#

#= (x-alpha)(x-beta)(x-gamma)#

#= x^3-(alpha+beta+gamma)x^2+(alphabeta+betagamma+gammaalpha)x-alphabetagamma#

We find:

#{ (alpha+beta+gamma=0), (alphabeta+betagamma+gammaalpha = -7), (alphabetagamma = 4) :}#

So:

#alpha^2+beta^2+gamma^2#

#= (alpha+beta+gamma)^2-2(alphabeta+betagamma+gammaalpha)= 0-2(-7) = 14#

#alpha^2beta^2+beta^2gamma^2+gamma^2alpha^2#

#=(alphabeta+betagamma+gammaalpha)^2-2alphabetagamma(alpha+beta+gamma) = (-7)^2-2(4)(0) = 49#

#alpha^4+beta^4+gamma^4#

#=(alpha^2+beta^2+gamma^2)^2-2(alpha^2beta^2+beta^2gamma^2+gamma^2alpha^2) = 14^2-2(49) = 196-98 = 98#

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