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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