α , β , γ are roots of cubic equation x 3 + 4 x − 1 = 0

Question

α  ,  β  ,  γ are roots of cubic equation       x    3    +  4  x  −  1  =  0

Samantha Braun · Accepted Answer

Your calculation leads us                    (        β        +        1        )        (        γ        +        1        )                    α        2              +                  (        γ        +        1        )        (        α        +        1        )                    β        2              +                  (        α        +        1        )        (        β        +        1        )                    γ        2              =  6      {          1              (        α        +        1        )                  α          2                      +          1              (        β        +        1        )                  β          2                      +          1              (        γ        +        1        )                  γ          2                      }    .Here, by Vieta&#039;s formulas, notice that      7    6    =      1          α      +      1        +      1          β      +      1        +      1          γ      +      1        =            (      α      β      +      β      γ      +      γ      α      )      +      2      (      α      +      β      +      γ      )      +      3              (      α      +      1      )      (      β      +      1      )      (      γ      +      1      )        .Since   α  +  β  +  γ  =  0  , we have  (  α  +  1  )  (  β  +  1  )  (  γ  +  1  )  =  6.Noting that your last   {  } equals to            (                        α                3            +                        β                3            +                        γ                3            )      +      (                        α                2            +                        β                2            +                        γ                2            )              (      α      β      γ              )        2            (      α      +      1      )      (      β      +      1      )      (      γ      +      1      )        ,we can use            α        3    +            β        3    +            γ        3    =  3  α  β  γ  +  (  α  +  β  +  γ  )  (            α        2    +            β        2    +            γ        2    −  α  β  −  β  γ  −  γ  α  )  .with  α  +  β  +  γ  =  0  ,  α  β  +  β  γ  +  γ  α  =  4  ,  α  β  γ  =  1and            α        2    +            β        2    +            γ        2    =  (  α  +  β  +  γ      )    2    −  2  (  α  β  +  β  γ  +  γ  α  )  .Now you&#039;ll be able to get the answer.

alpha, beta, gamma are roots of cubic equation x^3+4x−1=0

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