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Let $z_1,z_2,z_3$ be three complex number such that :

$| z_1| = | z_2| = | z_3| = 1$ and $z^3_1 + z^3_2 + z^3_3 +z_1z_2z_3=0$

Then $|z_1+z_2+z_3|$ can take which of the following values ?

(A) $1$ (B) $2$ (C) $3$ (D) $4$

Using the triangle inequality I get $|z_1+z_2+z_3| \leq |z_1|+ |z_2| + |z_3| \leq 3$.

Using the inequality the answers should be (A), (B) and (C).

However, my book says that the answer is only (A) and (B). What is wrong in my method?

  • 0
    You have only shown that the number is less than or equal to 3, not that there actually exists $z_1, z_2, z_3$ such that it equals 3. – 2017-02-17
  • 0
    For starters, in your reasoning you did not use the second equality=) – 2017-02-17

1 Answers 1

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Note that in order for $|z_1+z_2+z_3|$ to be equal to $3$, you need that all $z_i$ must be equal , i.e. $z_1=z_2=z_3$, which contradicts the second equality (it would become $4z_1^3 =0$, with the only solution $z_1=0$).

Now you need to show that (A) and (B) are valid answers.