Where's the First Powers?

Algebra Level 5

Given that the complex numbers \(x,y,z\) satisfy the equations

\[ \begin{eqnarray} x^2 + y^2 + z^2 & = & 2 \\ x^3 + y^3 + z^3 & = & 3 \\ x^4 + y^4 + z^4 & = & 4 \\ \end{eqnarray} \]

Denote \( f \) as a minimum degree monic polynomial with real coefficients such that it has root \(x+y+z\).

What is the value of \(f(1)\)?

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