Closer to your roots

Consider the polynomial $f(x) = x^7 - 4x^3 + x + 1$ . All the zeroes of the above polynomial are plotted in the complex plane i.e., the argand plane. How many are within a unit distance from the origin?

Details and Assumptions:

• The repeated roots are counted with multiplicity i.e., $(2x - 1)^2 = 0$ has two solutions within a unit distance from origin and not one.

• There is a very neat and simple way of doing without W|A. Please refrain from using any such mathematical computational softwares.