The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and $$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$Prove that $f$ has at least two complex non-real roots.
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Tags: algebra, functional
The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and $$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$Prove that $f$ has at least two complex non-real roots.