How to solve these three equations?If α , β , γ are three numbers s.t.: α + β + γ = − 2 α 2 + β 2 + γ 2 = 6 α 3 + β 3 + γ 3 = − 5 , then α 4 + β 4 + γ 4 is equal to ??

Question

How to solve these three equations?If       α    ,      β    ,      γ   are three numbers s.t.:      α    +      β    +      γ    =  −  2              α        2    +            β        2    +            γ        2    =  6              α        3    +            β        3    +            γ        3    =  −  5  ,  then             α        4    +            β        4    +            γ        4   is equal to ??

jarakapak7 · Accepted Answer

Let       A      n    =      a    n    +      b    n    +      c    n  . Then we have      A          n      +      3        =  (  a  +  b  +  c  )      A          n      +      2        −  (  a  b  +  b  c  +  a  c  )      A          n      +      1        +  a  b  c      A      n  and  2  (  a  b  +  b  c  +  a  c  )  =  (  a  +  b  +  c      )    2    −  (      a    2    +      b    2    +      c    2    )  =  4  −  6  =  −  2      a    3    +      b    3    +      c    3    =  (  a  +  b  +  c      )    3    −  3  (  a  +  b  +  c  )  (  a  b  +  b  c  +  a  c  )  +  3  a  b  cThen we can easily find   a  b  c.

How to solve these three equations? If <mrow class="MJX-TeXAtom-ORD"> α </mrow>

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