Algebra II

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

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

Thanks for your help.

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  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.

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