State the possible number of imaginary zeros of g(x)= x4 + 3x3 + 7x2 - 6x - 13.

Question

I came up with the answer of 2,4 or 0.
Is this correct?

Thanks for your help.

Answer 1

Substitute x = i t where t is assumed to be real:

t^4 - 3it^3 - 7t^2 -6 i t - 13 = 0

Take the imaginary part of the equation:

t^3 + 2 t = 0 --->

t(t^2 + 2) = 0

Now, we can see from the original equation for t that t = 0 is not a solution (this is actually implied by the real part of the equation). Since t has to be real, the equation:

t^2 + 2 = 0

cannot be satisfied.

Therefore the equation has no imaginary solutions.