Imaginary vs Complex roots If I'm asked whether a polynomial has any imaginary root, what does it mean?
Does it mean whether the polynomial has any root which is purely imaginary?
or
Does it mean whether the polynomial has any complex root, where real part is not zero?
 A: Both meanings are used (for instance, both are mentioned in the introduction of the Wikipedia page https://en.wikipedia.org/wiki/Imaginary_number).  So lacking any additional context, there is no way to know (though I would consider your first interpretation more likely).  You will have to figure out what makes sense from context or else ask whoever posed the question.
A: As complex number, we consider the number $z$, such that : 
$$z= x + yi$$
where $x,y \in \mathbb R$.
An imaginary number, is a number of the form above, but with $x=0$. More specifically, the number $z$ is considered imaginary, when : 
$$z=yi$$
where $y\in \mathbb R$. You may also come across that called as a purely imaginary number. 
The imaginary numbers lie on the vertical coordinate axis on the complex plane and thus the definition of a complex number stated above is derived, as real numbers lie on the horizontal axis.
The set of imaginary numbers is obviously a subset of the complex numbers : $\mathbb I \subset \mathbb C.$ The set of the real numbers is a subset of the complex numbers : $\mathbb R \subset \mathbb C$ since all the real numbers are derived if $y=0$.
Thus, if you're asked to determine whether a polynomial (or generally an equation) has imaginary roots, you should consider it as checking if it has purely imaginary roots, else you would be asked about complex roots.
There is an important differentiation between purely imaginary and complex on many fields of mathematics and one example is the type of a stationary point while discussing dynamical systems : If the eigenvalues of the matrix of the system/linearised-system are complex then the stationary point is a focus (with some properties regarding the complex number...) but when the eigenvalues are purely imaginary (or imaginary simply as one may write) then there's a different case, as the stationary point is then considered a center.
Exercise examples :
The roots of the equation $x^2 = -4$ are the purely imaginary numbers $x = \pm 2i$.
The roots of the equation $x^2+x+1=0$ are the complex numbers $x= -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$.
