SOLUTION: Show that the equation z^8 + (1 + i)z^3 + 2z^2 = 10 + 24i has no solution z ∈ C with |z| ≤ √2. Carefully explain your reasoning.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Show that the equation z^8 + (1 + i)z^3 + 2z^2 = 10 + 24i has no solution z ∈ C with |z| ≤ √2. Carefully explain your reasoning.      Log On


   

Question 1106748: Show that the equation
z^8 + (1 + i)z^3 + 2z^2 = 10 + 24i
has no solution z ∈ C with |z| ≤ √2.
Carefully explain your reasoning.
Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  Evaluate the modulus  of the left side for |z| <= sqrt%282%29:


       |z^8 + (1+i)*z^3 + 2*z^2| <= (apply the "triangle inequality" to the modulus of the sum of complex numbers) <=

    <= |z^8| + |(1+i)|*|z*3| + 2*|z^2| <= %28sqrt%282%29%29%5E8 + sqrt%282%29%2A%28sqrt%282%29%29%5E3 + 2%2A%28sqrt%282%29%29%5E2 = 16 + 4 + 4 = 24.



2.  Evaluate the modulus of the right side:

    |10 + 24i| = sqrt%2810%5E2+%2B+24%5E2%29 > 24.



3.  Comparing n.1  with n.2,  you may conclude that the equation  HAS NO solutions  in the given domain.


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