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I'll get straight down to it: I have a polynomial with a complex variable z and complex coeff's (real parts all zero): z3 + 8i z2 - z + 42i, i being the imaginary unit. I am wondering for which values of z does this polynomial give real numbers, that is numbers with no imaginary part. I know I can write the equation z3 + 8i z2 - z + 42i - r = 0 where r is a real number. However, I am looking for keywords to look up, techniques to read up on and (hopefully) learn, different kind of plots, to be able to plot r as z moves across the complex plane. The more suggestions the merrier. I know that there can be more than one z in the original polynomial that produces the same r. You might say I want to know at which points does the polynomial turn all real.
