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Root Magnitude Proof (Posted on 2025-02-08)

Prove that if 11z¹⁰ +10iz⁹ +10iz−11 = 0, then |z| = 1.

No Solution Yet Submitted by Danish Ahmed Khan

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Possible proof

| Comment 1 of 6

Divide by z^5. (valid if |z| = 1)

Factor the expression.

11z^5 + 10iz^4 +10iz^(-4) − 11z^(-5) = 0

11z^5 − 11z^(-5) + 10iz^4 +10iz^(-4) = 0

11(z^5 − z^(-5)) + 10i(z^4 + z^(-4)) = 0

Both terms must be zero (I think, not certain since z is complex).

z^5 = z^(-5)

z^10 = 1

z = the 10 roots of unity

z^4 + z^(-4) = 0

z^8 = -1

z = the 8 roots of -1

Thus |z| = 1

Posted by Larry on 2025-02-08 14:45:24

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