So I was fiddling with graphs and then it hit me what would a cubic polynomial with only complex roots look like ,i.e., no root which is purely real. I gave such an equation to Wolfram alpha which was
$$f(x) = (x-i)(x-2i)(x-3i)$$
My hypothesis was that the graph wouldn't exist in Cartesian system and only roots in complex plane would be shown. But Wolfram alpha expanded the expression and put the factored terms without i on one side and with $i$ on the other. I was able to understand that it was showing me the graphs of real and complex parts on the Cartesian plane but I was not able to understand its utility. Like what is its purpose when the roots would still remain the same. Moreover, the output of real and imaginary parts was same even for $4$ degree and higher polynomials when to my understanding such graphs should be able to be completely above the x-axis for even powers of $x$. Can someone explain to me what are their utilities. Also I noticed that both complex and real parts were equations of the form where the powers were even numbers with difference of $2$. Such as,
$$ax⁶ + bx⁴ + cx² + d$$
Is this something that happens often or is it just the bias of my polynomial.
Edit:
Links for the outputs:
Link 1: https://www.wolframalpha.com/input?i=%28x-i%29%28x-2i%29%28x-3i%29
This is the case of cubic polynomial where the roots are completely imaginary and there has been factorisation done in a column assuming x as real and then two graphs have been plotted.
Link 2:
https://www.wolframalpha.com/input?i=%28x-i%29%28x-2i%29%28x%2B3i%29%28x-4i%29
Similar result was observed for quartic and upper polynomials.
Also for the graph I realised I had given it the prompt of (x-i)(x+i) which resolved into x²+1 and as such a graph was possible for it. So I got the answer for that part.