Proving that a polynomial has no roots in the unit circle

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I want to prove that if $|z|=1 $ then $z^8-3z^2+1 neq 0$. I tried to prove the reciprocal by taking norms in $z^8-3z^2+1= 0$ and then solving for $ |z|$ but it does not work. I also assume $| z|=1 $ and then trying to see that $| z^8-3z^2+1 |> 0 $ but it did not work neither.



Any ideas on this?






complex-analysis polynomials roots







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    up vote
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    I want to prove that if $|z|=1 $ then $z^8-3z^2+1 neq 0$. I tried to prove the reciprocal by taking norms in $z^8-3z^2+1= 0$ and then solving for $ |z|$ but it does not work. I also assume $| z|=1 $ and then trying to see that $| z^8-3z^2+1 |> 0 $ but it did not work neither.



    Any ideas on this?






    complex-analysis polynomials roots







    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I want to prove that if $|z|=1 $ then $z^8-3z^2+1 neq 0$. I tried to prove the reciprocal by taking norms in $z^8-3z^2+1= 0$ and then solving for $ |z|$ but it does not work. I also assume $| z|=1 $ and then trying to see that $| z^8-3z^2+1 |> 0 $ but it did not work neither.



      Any ideas on this?






      complex-analysis polynomials roots







      share|cite|improve this question













      I want to prove that if $|z|=1 $ then $z^8-3z^2+1 neq 0$. I tried to prove the reciprocal by taking norms in $z^8-3z^2+1= 0$ and then solving for $ |z|$ but it does not work. I also assume $| z|=1 $ and then trying to see that $| z^8-3z^2+1 |> 0 $ but it did not work neither.



      Any ideas on this?






      complex-analysis polynomials roots




      complex-analysis polynomials roots






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      share|cite|improve this question











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      asked 2 hours ago









      Natalio

      1749




      1749




















          3 Answers
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          If $z^8-3z^2+1 = 0$ then
          $$
          3 |z|^2 = |3 z^2| = |z^8 + 1| le |z|^8 + 1
          $$
          and that is not possible if $|z| = 1$.






          share|cite|improve this answer



























            up vote
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            $$|z|=1 implies |z^8|=1 $$



            Therefore, $|3z^2-1|=|z^8|=1$



            Note that $|3z^2-1| ge ||3z^2|-1| =2$



            Which is not consistent with $|3z^2-1|=1$






            share|cite|improve this answer



























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              down vote













              Try the reverse triange inequality:
              $$|z^8 - 3z^2 + 1| ge |3z^2| - |z^8| - 1.$$






              share|cite|improve this answer




















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                3 Answers
                3






                active

                oldest

                votes








                3 Answers
                3






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes








                up vote
                4
                down vote



                accepted










                If $z^8-3z^2+1 = 0$ then
                $$
                3 |z|^2 = |3 z^2| = |z^8 + 1| le |z|^8 + 1
                $$
                and that is not possible if $|z| = 1$.






                share|cite|improve this answer
























                  up vote
                  4
                  down vote



                  accepted










                  If $z^8-3z^2+1 = 0$ then
                  $$
                  3 |z|^2 = |3 z^2| = |z^8 + 1| le |z|^8 + 1
                  $$
                  and that is not possible if $|z| = 1$.






                  share|cite|improve this answer






















                    up vote
                    4
                    down vote



                    accepted







                    up vote
                    4
                    down vote



                    accepted






                    If $z^8-3z^2+1 = 0$ then
                    $$
                    3 |z|^2 = |3 z^2| = |z^8 + 1| le |z|^8 + 1
                    $$
                    and that is not possible if $|z| = 1$.






                    share|cite|improve this answer












                    If $z^8-3z^2+1 = 0$ then
                    $$
                    3 |z|^2 = |3 z^2| = |z^8 + 1| le |z|^8 + 1
                    $$
                    and that is not possible if $|z| = 1$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 hours ago









                    Martin R

                    24.4k32844




                    24.4k32844




















                        up vote
                        1
                        down vote













                        $$|z|=1 implies |z^8|=1 $$



                        Therefore, $|3z^2-1|=|z^8|=1$



                        Note that $|3z^2-1| ge ||3z^2|-1| =2$



                        Which is not consistent with $|3z^2-1|=1$






                        share|cite|improve this answer
























                          up vote
                          1
                          down vote













                          $$|z|=1 implies |z^8|=1 $$



                          Therefore, $|3z^2-1|=|z^8|=1$



                          Note that $|3z^2-1| ge ||3z^2|-1| =2$



                          Which is not consistent with $|3z^2-1|=1$






                          share|cite|improve this answer






















                            up vote
                            1
                            down vote










                            up vote
                            1
                            down vote









                            $$|z|=1 implies |z^8|=1 $$



                            Therefore, $|3z^2-1|=|z^8|=1$



                            Note that $|3z^2-1| ge ||3z^2|-1| =2$



                            Which is not consistent with $|3z^2-1|=1$






                            share|cite|improve this answer












                            $$|z|=1 implies |z^8|=1 $$



                            Therefore, $|3z^2-1|=|z^8|=1$



                            Note that $|3z^2-1| ge ||3z^2|-1| =2$



                            Which is not consistent with $|3z^2-1|=1$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 2 hours ago









                            Mohammad Riazi-Kermani

                            33.4k41855




                            33.4k41855




















                                up vote
                                0
                                down vote













                                Try the reverse triange inequality:
                                $$|z^8 - 3z^2 + 1| ge |3z^2| - |z^8| - 1.$$






                                share|cite|improve this answer
























                                  up vote
                                  0
                                  down vote













                                  Try the reverse triange inequality:
                                  $$|z^8 - 3z^2 + 1| ge |3z^2| - |z^8| - 1.$$






                                  share|cite|improve this answer






















                                    up vote
                                    0
                                    down vote










                                    up vote
                                    0
                                    down vote









                                    Try the reverse triange inequality:
                                    $$|z^8 - 3z^2 + 1| ge |3z^2| - |z^8| - 1.$$






                                    share|cite|improve this answer












                                    Try the reverse triange inequality:
                                    $$|z^8 - 3z^2 + 1| ge |3z^2| - |z^8| - 1.$$







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 2 hours ago









                                    Umberto P.

                                    35.8k12860




                                    35.8k12860



























                                         

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