Evaluating the modulus of roots of a 6 degree polynomial

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Let $x_1,x_2,x_3,x_4,x_5,x_6$ be the roots of the polynomial equation



$$x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0.$$



Then,



(A)$|x_i|=2$ for exactly one value of $i$



(B)$|x_i|=2$ for exactly two values of $i$



(C)$|x_i|=2$ for all values of $i$.



(D)$|x_i|=2$ for no value of $i$.



My attempt: I noticed that the polynomial forms a geometric



progression with common ratio $2/x$.



Then I summed the terms and found this relation $x^7=128$ which gives us



$x=2$ but $x=2$ doesn't satisfy the original polynomial.



I looked at the graph too and found that the polynomial has no real roots .



So now , how to find the modulus of the roots ?










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  • 1




    You found $x^ 7 = 128$. Note that $x neq 2$, of course. It turns out that, in fact, the above polynomial is a factor of $x^7 - 128$. Consequently, every root must have modulus equal to $2$, and we can write down all the roots explicitly using the seventh roots of unity.
    – Ð°ÑÑ‚он вілла олоф мэллбэрг
    Aug 21 at 12:21















up vote
5
down vote

favorite












Let $x_1,x_2,x_3,x_4,x_5,x_6$ be the roots of the polynomial equation



$$x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0.$$



Then,



(A)$|x_i|=2$ for exactly one value of $i$



(B)$|x_i|=2$ for exactly two values of $i$



(C)$|x_i|=2$ for all values of $i$.



(D)$|x_i|=2$ for no value of $i$.



My attempt: I noticed that the polynomial forms a geometric



progression with common ratio $2/x$.



Then I summed the terms and found this relation $x^7=128$ which gives us



$x=2$ but $x=2$ doesn't satisfy the original polynomial.



I looked at the graph too and found that the polynomial has no real roots .



So now , how to find the modulus of the roots ?










share|cite|improve this question



















  • 1




    You found $x^ 7 = 128$. Note that $x neq 2$, of course. It turns out that, in fact, the above polynomial is a factor of $x^7 - 128$. Consequently, every root must have modulus equal to $2$, and we can write down all the roots explicitly using the seventh roots of unity.
    – Ð°ÑÑ‚он вілла олоф мэллбэрг
    Aug 21 at 12:21













up vote
5
down vote

favorite









up vote
5
down vote

favorite











Let $x_1,x_2,x_3,x_4,x_5,x_6$ be the roots of the polynomial equation



$$x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0.$$



Then,



(A)$|x_i|=2$ for exactly one value of $i$



(B)$|x_i|=2$ for exactly two values of $i$



(C)$|x_i|=2$ for all values of $i$.



(D)$|x_i|=2$ for no value of $i$.



My attempt: I noticed that the polynomial forms a geometric



progression with common ratio $2/x$.



Then I summed the terms and found this relation $x^7=128$ which gives us



$x=2$ but $x=2$ doesn't satisfy the original polynomial.



I looked at the graph too and found that the polynomial has no real roots .



So now , how to find the modulus of the roots ?










share|cite|improve this question















Let $x_1,x_2,x_3,x_4,x_5,x_6$ be the roots of the polynomial equation



$$x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0.$$



Then,



(A)$|x_i|=2$ for exactly one value of $i$



(B)$|x_i|=2$ for exactly two values of $i$



(C)$|x_i|=2$ for all values of $i$.



(D)$|x_i|=2$ for no value of $i$.



My attempt: I noticed that the polynomial forms a geometric



progression with common ratio $2/x$.



Then I summed the terms and found this relation $x^7=128$ which gives us



$x=2$ but $x=2$ doesn't satisfy the original polynomial.



I looked at the graph too and found that the polynomial has no real roots .



So now , how to find the modulus of the roots ?







polynomials roots






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edited Aug 21 at 12:18









Clayton

18.3k22883




18.3k22883










asked Aug 21 at 12:15









Alphanerd

8719




8719







  • 1




    You found $x^ 7 = 128$. Note that $x neq 2$, of course. It turns out that, in fact, the above polynomial is a factor of $x^7 - 128$. Consequently, every root must have modulus equal to $2$, and we can write down all the roots explicitly using the seventh roots of unity.
    – Ð°ÑÑ‚он вілла олоф мэллбэрг
    Aug 21 at 12:21













  • 1




    You found $x^ 7 = 128$. Note that $x neq 2$, of course. It turns out that, in fact, the above polynomial is a factor of $x^7 - 128$. Consequently, every root must have modulus equal to $2$, and we can write down all the roots explicitly using the seventh roots of unity.
    – Ð°ÑÑ‚он вілла олоф мэллбэрг
    Aug 21 at 12:21








1




1




You found $x^ 7 = 128$. Note that $x neq 2$, of course. It turns out that, in fact, the above polynomial is a factor of $x^7 - 128$. Consequently, every root must have modulus equal to $2$, and we can write down all the roots explicitly using the seventh roots of unity.
– Ð°ÑÑ‚он вілла олоф мэллбэрг
Aug 21 at 12:21





You found $x^ 7 = 128$. Note that $x neq 2$, of course. It turns out that, in fact, the above polynomial is a factor of $x^7 - 128$. Consequently, every root must have modulus equal to $2$, and we can write down all the roots explicitly using the seventh roots of unity.
– Ð°ÑÑ‚он вілла олоф мэллбэрг
Aug 21 at 12:21











2 Answers
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$$f(x)=sum_r=0^6(x^r2^6-r)=2^6dfracleft(dfrac x2right)^7-1dfrac x2-1=dfracx^7-128x-2$$



Clearly, $f(2)ne0$



$implies x^7=128$ with $xne2$



$iffleft(dfrac x2right)^7=1$



Now take modulus in both sides using $|z^m|=|z|^m$






share|cite|improve this answer



























    up vote
    3
    down vote













    As $x=2$ is not a root, the equation is



    $$fracx^7-128x-2=0.$$



    From this we draw



    $$|x^7|=|x|^7=128$$ and the only real solution $$|x|=2.$$






    share|cite|improve this answer




















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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote



      accepted










      $$f(x)=sum_r=0^6(x^r2^6-r)=2^6dfracleft(dfrac x2right)^7-1dfrac x2-1=dfracx^7-128x-2$$



      Clearly, $f(2)ne0$



      $implies x^7=128$ with $xne2$



      $iffleft(dfrac x2right)^7=1$



      Now take modulus in both sides using $|z^m|=|z|^m$






      share|cite|improve this answer
























        up vote
        4
        down vote



        accepted










        $$f(x)=sum_r=0^6(x^r2^6-r)=2^6dfracleft(dfrac x2right)^7-1dfrac x2-1=dfracx^7-128x-2$$



        Clearly, $f(2)ne0$



        $implies x^7=128$ with $xne2$



        $iffleft(dfrac x2right)^7=1$



        Now take modulus in both sides using $|z^m|=|z|^m$






        share|cite|improve this answer






















          up vote
          4
          down vote



          accepted







          up vote
          4
          down vote



          accepted






          $$f(x)=sum_r=0^6(x^r2^6-r)=2^6dfracleft(dfrac x2right)^7-1dfrac x2-1=dfracx^7-128x-2$$



          Clearly, $f(2)ne0$



          $implies x^7=128$ with $xne2$



          $iffleft(dfrac x2right)^7=1$



          Now take modulus in both sides using $|z^m|=|z|^m$






          share|cite|improve this answer












          $$f(x)=sum_r=0^6(x^r2^6-r)=2^6dfracleft(dfrac x2right)^7-1dfrac x2-1=dfracx^7-128x-2$$



          Clearly, $f(2)ne0$



          $implies x^7=128$ with $xne2$



          $iffleft(dfrac x2right)^7=1$



          Now take modulus in both sides using $|z^m|=|z|^m$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 21 at 12:19









          lab bhattacharjee

          216k14153266




          216k14153266




















              up vote
              3
              down vote













              As $x=2$ is not a root, the equation is



              $$fracx^7-128x-2=0.$$



              From this we draw



              $$|x^7|=|x|^7=128$$ and the only real solution $$|x|=2.$$






              share|cite|improve this answer
























                up vote
                3
                down vote













                As $x=2$ is not a root, the equation is



                $$fracx^7-128x-2=0.$$



                From this we draw



                $$|x^7|=|x|^7=128$$ and the only real solution $$|x|=2.$$






                share|cite|improve this answer






















                  up vote
                  3
                  down vote










                  up vote
                  3
                  down vote









                  As $x=2$ is not a root, the equation is



                  $$fracx^7-128x-2=0.$$



                  From this we draw



                  $$|x^7|=|x|^7=128$$ and the only real solution $$|x|=2.$$






                  share|cite|improve this answer












                  As $x=2$ is not a root, the equation is



                  $$fracx^7-128x-2=0.$$



                  From this we draw



                  $$|x^7|=|x|^7=128$$ and the only real solution $$|x|=2.$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Aug 21 at 12:20









                  Yves Daoust

                  115k666209




                  115k666209



























                       

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