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  • It did not happen with the polynomial in your OP. But it does happen e.g. with the polynomial $$(x-2\cos\frac{2\pi}7)(x-2\cos\frac{4\pi}7)(x-2\cos\frac{8\pi}7)=x^3+x^2-2x+1$$??????(unless I made a mistake), because $\cos 2\alpha=2\cos^2\alpha-1$ is polynomially related with $\cos\alpha$.
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