Question

how many imaginary and real zeros does this function have?
f(x) = x3 - 20x2 + 123x - 216

Answer 1

Without a graphing calculator you could use the rational roots theorem to determine there are 3 rational roots. This requires a bit of guessing and checking though. But it will tell you there are 3 rational roots. Because this polynomial is of degree 3, that means there are at most 3 roots anyway. So all 3 roots are rational, and hence real.

Answer 2

Calculation software indicates that the expression can be factored as follows:
(x-9) (x-8) (x-3)

Answer 3

A plot is attached.