Almost pentagram

Algebra Level 4

If $z_1 , z_2 , z_3 , z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z^1 + 1 = 0$ , then what is the value of $\left | \displaystyle\sum_{i=1}^4 z_i ^4 \right |$ ?

The answer is 1.

1 solution

Nishant Rai
Apr 11, 2015

The given equation is $\frac {z^5 -1}{z - 1} = 0$ which means that $z_1 , z_2 , z_3 , z_4$ are four out of the five fifth roots of unity except 1.

$z_1^4 + z_2^4 + z_3^4 + z_4^4 +1^4 = 0$

$|\displaystyle\sum_{i=1}^4 z_i ^4| = 1$

Almost pentagram

The answer is 1.

1 solution

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