.
Consider .
By finding the roots in form, and using appropriate substitutions, show that
= 0.
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Equation = 0 is the same as = i.
One root is, obviously, z = i, since = i.
Let's list all the roots
= = ,
= = = = = i,
(we just noticed it above !)
= = = ,
= = = ,
= = = .
Notice that and have opposite real parts and identical imaginary parts. (*)
Similarly, and have opposite real parts and identical imaginary parts. (**)
We can write the decomposition of in the form of the product of linear binomials with the roots
= =
= . (1)
In this decomposition (1), second and third parentheses will give the product
= . (2)
Here = , as we noticed in (*), and = = = -1.
Therefore,
= .
Similarly, in decomposition (1), fourth and fifth parentheses will give the product
= . (3)
Here = , as we noticed in (**), and = = = -1.
Therefore,
= . (4)
Thus, combining everything in one piece, we get
If = 0, then = = 0.
QED.
At this point, the proof is complete.
Solved.
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In her post, @MathLover1 incorrectly read the problem and incorrectly understood
what the problem requested to prove.
So, her writing in her post is not a proof of the problem' statement
and has nothing in common with what this problem requests to prove.
For the peace in your mind, simply ignore that post.