Can -5, -8i, 4i, and 5 all be zero's of a 4th degree polynomial?

Question

Pre calculus. Zero's of polynomials

Answer 1

No, because when the polynomial is constructed from its factors there will be some imaginary components caused by the non-cancellation of terms involving i (sqrt(-1)), e.g., (x+8i)(x-4i)=x^2+4ix+32. The remaining factor is x^2-25, so there would be 4ix^3 and -100ix. Polynomials by definition are not complex.