a, b, c are three complex numbers such that roots of P ( x ) = x 3 + a x 2 + b x + c lie on a unit circle prove that roots of Q ( x ) = x 3 + | a | x 2 + | b | x + | c | lie on unit circle too.

Question

a, b, c are three complex numbers such that roots of   P  (  x  )  =      x    3    +  a      x    2    +  b  x  +  c lie on a unit circle prove that roots of   Q  (  x  )  =      x    3    +      |    a      |        x    2    +      |    b      |    x  +      |    c      |   lie on unit circle too.

Mihevcekd · Accepted Answer

We have      x    1    +      x    2    +      x    3    =  −  a  ,      x    1        x    2    +      x    2        x    3    +      x    3        x    1    =  b  ,      x    1        x    2        x    3    =  −  cIt is easy to observe that       |    c      |    =  1 and      |    b            |        2    =  1  +  1  +  1  +      x    1              x      2        ¯    +      x    2              x      1        ¯    +      x    1              x      3        ¯    +      x    3              x      1        ¯    +      x    3              x      2        ¯    +      x    2              x      3        ¯    =      |    a            |        2  thus  Q  (  x  )  =      x    3    +      |    a      |        x    2    +      |    a      |    x  +  1  =  (  x  +  1  )  (      x    2    −  x  +  1  +      |    a      |    x  )  =  (  x  +  1  )  (      x    2    −  (  1  −      |    a      |    )  x  +  1  )  .Denote   Δ  =  (  1  −      |    a      |        )    2    −  4  =  (      |    a      |    −  3  )  (      |    a      |    +  1  )  ≤  0. Assume that   Δ  &amp;lt;  0 then we have      x    1    =            1      −              |            a              |            −      i              −        Δ              2    ,      x    2    =            1      −              |            a              |            +      i              −        Δ              2  are also solutions of   Q  (  x  )  =  0 and      |        x    1              |        2    =      |        x    2              |        2    =            1      −      2              |            a              |            +              |            a                        |                2            −              |            a                        |                2            +      2              |            a              |            +      3        4    =  1therefore   Q  (  x  ) has three roots   (  −  1  ,      x    1    ,      x    2    ) on the unit circle.If   Δ  =  0 then   Q  (  x  )  =  (  x  +  1      )    3   and it is also true that Q has all roots on unit circle.

a, b, c are three complex numbers such that roots of P ( x ) = x 3 </msup>

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