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AP EAMCET2015MathematicsComplex Number

The common roots of the equations z^3+2 z^2+2 z+1=0, z²⁰¹⁴+z²⁰¹⁵+1=0 are

Options

  1. A, ^2
  2. B1, , ^2
  3. C-1, , ^2
  4. D- ,- ^2

Correct answer

A. , ^2

Step-by-step solution

Given equation, aligned & z^3+2 z^2+2 z+1=0 & (z+1) (z^2+z+1 )=0 aligned Its roots are -1, and ^2 . Let f(z)=z²⁰¹⁴+z²⁰¹⁵+1=0 Put z=-1, and ^2 respectively, we get f(-1)=(-1)²⁰¹⁴+(-1)²⁰¹⁵+1=0=1 0 Therefore, -1 is not a root of the equation f(z)=0 aligned & Again, f( )=( )²⁰¹⁴+( )²⁰¹⁵+1 & = ( ^3 )⁶⁷¹ + ( ^3 )⁶⁷¹ ^2+1 & = + ^2+1 & = ^2+ +1 & =0 aligned Therefore, is a root of the equation f(z)=0 Similarly, aligned & f ( ^3 )= ( ^2 )²⁰¹⁴+ ( ^3 )²⁰¹⁵+1 & = ( ^3 )¹³⁴² ^2+ ( ^3 )¹³⁴³ +1 & = ^2+ +1 & =0 aligned Hence and ^

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