If #iz^3 + z^2 -z + i = 0#, then prove that #|z| = 1#?

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Answer 1 · 2018-02-03T10:28:57

Kindly refer to a Proof in the Explanation.

Given that, #iz^3+z^2-z+i=0#,

#:. ul(iz^3+z^2)+ul(i^2z+i)=0............[because, i^2=-1]#,

#:. z^2(iz+1)+i(iz+1)=0#.

#:. (iz+1)(z^2+i)=0#.

#:. iz+1=0, or, z^2+i=0#.

#:. iz=-1, or, z^2=-i#.

#:. |iz|=|-1|=1, or, |z^2|=|-i|=1#.

#:. |i||z|=1, or, |z|^2=1...[because, |z_1z_2|=|z_1|||z_2|#.

#:. 1*|z|=1, or, |z|=+1......[because, |z|ge0]#.

Evidently, #|z|=1#.