# 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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