# How do you find all the zeros of #f(x)= -4x^8-9x^7+9x-4# with its multiplicities?

##### 1 Answer

See explanation...

#### Explanation:

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...

Then:

#f(iz) = -4(iz)^8-9(iz)^7+9(iz)-4#

#=-4z^8+9iz^7+9iz-4#

#=z^4(-4z^4+9iz^3+9iz^(-3)-4z^(-4))#

Let

#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: