Prove that cos ⁡ π 7 , cos ⁡ 3 π 7 , cos ⁡ 5 π 7 roots of polynomial 8 x 3 − 4 x 2 − 4 x + 1 = 0

Question

Prove that  cos  ⁡      π    7    ,  cos  ⁡            3      π        7    ,  cos  ⁡            5      π        7  roots of polynomial   8      x    3    −  4      x    2    −  4  x  +  1  =  0

overnachtt9xyx · Accepted Answer

Let   z  =      e                            i          π                7            . Then   cos  ⁡  (      π    7    )  =            z      +              z                  −          1                      2  ,   cos  ⁡  (            2      π        7    )  =                    z        2            +              z                  −          2                      2  ,   cos  ⁡  (            3      π        7    )  =                    z        3            +              z                  −          3                      2  . Also   0  =                    z        7            −      1              z      −      1        =  1  +  z  +      z    2    +      z    3    +      z    4    +      z    5    +      z    6  . As   z  ≠  0  ,, divide both sides by       z    3   to get      z    3    +      1          z      3        +      z    2    +      1          z      2        +  z  +      1    z    +  1  =  0.Now use            z      2        +          1              z        2              =                  (        z        +                  1          z                )            2        −    2    ,              z      3        +          1              z        3              =                  (        z        +                  1          z                )            3        −    3          (      z      +              1        z            )      to obtain a cubic polynomial with the above roots

Prove that cos ⁡ π 7 </mfrac> , co

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