how do you know if theres an imaginary zero on a f(x)?

Question

ex: f(x) = x^3 - 2x^2 + x - 2

Answer 1

x^2(x-2)+(x-2)=0
(x-2)(x^2+1)=0 then x-2=0=> x=2 is real solution and:x^2+1=0which a x=+/-i the imaginary zero.

Answer 2

The first step is to factor the equation down to factors of no bigger than 2.

f(x)=x^3-2x^2+x-2
=x^2(x-2)+x-2
=(x^2+1)(x-2)

Then set the equation equal to zero, and find all possible answers

In this case, x must equal 2 or x^2 must equal -1, which is impossible, so there is at least one imaginary zero

Answer 3

Factor it.

f(x) = x^3 - 2x^2 + x - 2 = (x - 2)(x^2 +1) = 0
The only real solution is x = 2.
Clearly x^2 + 1 = 0 will only have imaginary solutions.