Doppelgangers

Algebra Level 4

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

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

0=p(z)p(zˉ)? 0=p(z) \neq p(\bar{z}) ?

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