If α , β , γ are roots of the cubic equation 2 x 3 + 3 x 2 − x − 1 = 0

Question

If   α  ,  β  ,  γ  are roots of the cubic equation   2      x          3        +  3      x    2    −  x  −  1  =  0

Aaron Everett · Accepted Answer

If  2      x    3    +  3      x    2    −  x  −  1  =  0has roots   α  ,  β  ,  γ, then substituting   x  ↦      1    x   (and multiplying by   −      x    3   to clear denominators)                                  −                  x          3                          (                      2                          x              3                                +                      3                          x              2                                −                      1            x                    −          1          )                                                  =                  x          3                +                  x          2                −        3        x        −        2        =        0            has roots       1    α    ,      1    β    ,      1    γ  . Then substituting   x  ↦  −      2    3    x (and multiplying by   −      27    8   to clear denominators)                                  −                  27          8                          (                                    (              −                              2                3                            x              )                        3                    +                                    (              −                              2                3                            x              )                        2                    −          3                      (            −                          2              3                        x            )                    −          2          )                                                    x          3                −                  3          2                          x          2                −                  27          4                x        +                  27          4                =        0            has roots             −      3              /            2        α    =            α      +      β      +      γ        α    ,            −      3              /            2        β    =            α      +      β      +      γ        β    ,            −      3              /            2        γ    =            α      +      β      +      γ        γ  . Next, substituting   x  ↦  x  +  1                                  (        x        +        1                  )          3                −                  3          2                (        x        +        1                  )          2                −                  27          4                (        x        +        1        )        +                  27          4                                                  =                  x          3                +                  3          2                          x          2                −                  27          4                x        −                  1          2                =        0            has roots             β      +      γ        α    ,            α      +      γ        β    ,            α      +      β        γ  . Finally, substituting   x  ↦      1    x   (and multiplying by   −  4      x    3   to clear denominators)                                  −        4                  x          3                          (                      1                          x              3                                +                      3            2                                1                          x              2                                −                      27            4                                1            x                    −                      1            2                    )                                                  =        2                  x          3                +        27                  x          2                −        6        x        −        4        =        0            has roots       α          β      +      γ        ,      β          α      +      γ        ,      γ          α      +      β

If α , β , γ are roots of the cubic

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