Doppelgangers

Algebra Level 3

Consider the function defined as $p(x) = \sum_{i=1}^{n} \alpha_i x^{\beta_i}$ where, $$n$$ is a positive integer and $$\alpha_i,\beta_i$$ are real numbers.

Is it possible that a complex number $$z$$ and its complex conjugate $$\bar{z}$$ exists such that

$0=p(z) \neq p(\bar{z}) ?$

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