find the imaginary solutions of the equation by factoring. x to the third power - 16x = 0

x^3 - 16x = 0

x(x^2 - 16) = 0
x(x+4)(x-4) = 0
x = 0 , 4 , -4

there are 3 real roots, none imaginary.

To find the imaginary solutions of the equation x^3 - 16x = 0 by factoring, we can start by factoring out the greatest common factor, which in this case is x:

x(x^2 - 16) = 0

Next, we can factor the quadratic expression x^2 - 16. This can be factored as a difference of squares:

x(x - 4)(x + 4) = 0

Now, we can set each factor equal to zero and solve for x:

x = 0
x - 4 = 0 => x = 4
x + 4 = 0 => x = -4

Therefore, the solutions to the equation x^3 - 16x = 0 are x = 0, x = 4, and x = -4.

Note: It's worth mentioning that the equation does not have any imaginary solutions because all the solutions are real numbers.