if a+b+c=0 then prove 1/x^b+x^-c+1+1/x^c+x^-a+1+1/x^a+x^-b+1=1​

Question

if a+b+c=0 then prove 1/x^b+x^-c+1+1/x^c+x^-a+1+1/x^a+x^-b+1=1​

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Skylar 6 days 2021-10-09T04:46:07+00:00 1 Answer 0 views 0

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    2021-10-09T04:47:35+00:00

    Answer:

    ANSWER

    Given

    a+b+c=0

    a+b=−c

    Cubing on both sides

    (a+b)

    3

    =(−c)

    3

    a

    3

    +b

    3

    +3(ab)(a+b)=−c

    3

    a

    3

    +b

    3

    +3ab(−c)=c

    3

    a

    3

    +b

    3

    −3abc=−c

    3

    a

    3

    +b

    3

    +c

    3

    =3abc

    =

    3ab

    (a+b)

    2

    +

    3bc

    (−a)

    2

    +

    3ac

    (−b)

    2

    =

    3ab

    (−c)

    2

    +

    3bc

    (−a)

    2

    +

    3ac

    (−b)

    2

    =

    3ab

    c

    2

    +

    3bc

    a

    2

    +

    3ac

    b

    2

    =

    3abc

    c

    3

    +a

    3

    +b

    3

    =

    3abc

    3abc

    =1

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