Where's the First Powers?

Algebra Level 5

Given that the complex numbers x,y,zx,y,z satisfy the equations

x2+y2+z2=2x3+y3+z3=3x4+y4+z4=4 \begin{aligned} x^2 + y^2 + z^2 & = & 2 \\ x^3 + y^3 + z^3 & = & 3 \\ x^4 + y^4 + z^4 & = & 4 \\ \end{aligned}

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

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

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