Will you find the value of roots?

Geometry Level 5

If the cubic polynomial whose roots are \( \cos \left( \dfrac{2\pi}7 \right),\cos \left( \dfrac{4\pi}7 \right)\) and \( \cos \left( \dfrac{6\pi}7 \right)\) is of the form \[ a^b x^3 + a^c x^2 - a^d x - 1 \; , \] where \(a,b,c\) and \(d\) are primes, find \(a+b+c+d\).

Hint:

\[{z^{2n+1}-1 =(z-1) \cdot \prod_{k=1}^{n}\left(z^{2}-2 \cdot \cos \dfrac{2\pi k}{2n+1}z +1 \right) }\]

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