How do you find all the zeros of #f(x)= -4x^8-9x^7+9x-4# with its multiplicities?
This seems a strangely difficult problem to be given, but we can at least reduce it from an octic to a quartic, using the semi-symmetry of the coefficients...
#f(iz) = -4(iz)^8-9(iz)^7+9(iz)-4#
#f(iz)/z^4 = -4t^4+9it^3+16t^2-27it-8#
This quartic is not very simple to solve, but it is at least possible to solve algebraically. Each zero
For example, one of the zeros of this quartic is: