How we can solve the equation $z^3+3z+2i=0$ ? And is there exist a general method to solve similar equation?
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You could notice that $z=-i$ is a solution. Then, find the quadratic and get the roots. – Claude Leibovici Oct 20 '14 at 07:53
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1@ClaudeLeibovici You mean $z=-i$, no? – David H Oct 20 '14 at 07:54
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One way to do this, since $z$ appears only to odd powers, is to set $z=iw$ which enables $i$ to be cancelled and gives you integer coefficients.
Another way is to spot a solution.
The general methods for solving a cubic also work with complex coefficients.
Mark Bennet
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Thank you Mr.Bennet. What is that method with complex coefficients? – SKMohammadi Oct 20 '14 at 08:07
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Here is my solution: We have $$z^3+3z+2i=z(z^2+1)+2(z+i)=z(z+i)(z-i)+2(z+i)=(z+i)(z^2-iz+2)=(z+i)^2(z-2i)=0$$ so our equation has three roots $-i,-i,2i$.
SKMohammadi
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