Problem finding zeros of complex polynomial z2+(3+i)|z|z―2=0

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Problem finding zeros of complex polynomial  z2+(3+i)|z|z―2=0

nick1337 · Accepted Answer

Step 1 The relation is equivalent to z2=−(3+i)|z|z¯2×z=0 is a solution, so in the following z≠0. Take modulus of both sides and denote r=|z|=|z¯|. Then r2=2r3, which means r=12 The relations turns to z2+12(3+i)z¯2=0 Multiply by z2 adn get z4+12(3+i)116=0 Write it in trigonometric form z4=116(−32−i12)=116(cos⁡7π6+sin⁡7π6) From here on it is just the extraction of complex roots. I did not answer your question, as to how to continue your calculations, but I can say from experience that in most complex numbers problems the substitution z=a+bi gets you in more troubles in the end, than working with the properties of complex conjugate, modulus and trigonometric form. You can see that in my solution no great computational problems were encountered.

Vasquez · Accepted Answer

Step 1Here is an alternative to solving it using polar form. Letz=a+biso thatz¯=a−biand|z|=a2+b2Then you want to solve(a+bi)2+(3+i)a2+b2(a−bi)2=0which expands to(a2−b2)+2abi+(3+i)a2+b2((a2−b2)−2abi)=0Thus, we need both the real part and the imaginary part of the left side to be 0, i.e.(a2−b2)+a2+b2(3×(a2−b2)+2ab)=0and2ab+a2+b2(−2ab3+(a2−b2))=0It should be possible to solve these equations by simple manipulations, though I haven&#039;t worked it out myself yet.

Problem finding zeros of complex polynomial z^{2}+(\sqrt{3}+i)|z|\bar{z}^{2}=0

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