Is the AREA imaginary or real?

Algebra Level 4

If the area of the polygon whose vertices are the solutions (in the complex plane) to the equation \({ x }^{ 7 }+{ x }^{ 6 }+{ x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 3 }+{ x }^{ 2 }+x+1=0\), can be expressed in the simplest form as \(\cfrac { \xi \sqrt { \vartheta } +\Im }{ \forall } \). Find the value of \(\xi +\vartheta +\Im +\forall \).

\(Details\)

\(\xi ,\vartheta ,\Im ,\forall \quad \quad \epsilon \quad Positive Integers\)

\(\vartheta \) is a square free positive integer.

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