Can a functions zeros be irrational and imaginary at the same time?

Question

This is the function: x^3 - x^2 - x - 2

how many complex zeros does this function have?

classify each zero as real, rational, irrational, or imaginary

Answer 1

A zero of a function f(x) occurs when the when f(r) = 0. r can be real including rational and irrational numbers or can be complex or pure imaginary. If complex or pure imaginary, the roots must come in pairs.

The function given is of the 3rd degree and hence has three zeroes. One is real and rational and equals 2.

The other two zeroes are complex and of the form a +bi where i = sqrt(-1).

a and b are real numbers. If b = 0, the a+bi is real. If a=0 then a+bi is pure imaginary. If a and b both not equal zero, the a+bi is complex. If a=bi is a zero the a-bi is also a zero
If 2i is a zero, then i2i is also a zero.

So 2 is a real rational complex zero.
The other two roots are imaginary.
Altogether there are three complex zeroes.

Answer 2

Your roots are:
x = 2 (real and rational)
x = -1/2 + (1/2 sqrt 3)i (imaginary and irrational)
x = -1/2 - (1/2 sqrt 3)i (imaginary and irrational)

Answer 3

Certainly a number such as the square root of negative 3 can be both irrational and imaginary.

I gave up solving those other than out of desperation 35 years ago.