Find the least possible number of imaginary roots of $x^9-x^5+x^4+x^2+1=0.$

Let. f{x) = x9 - x5 + x4 + x2 + 1

f{x): +-+++ 

f{x): -+ + + + 

There are 2 changes in. f(x) and 1 change in. f(-x). So j{x) has 2 +ve real roots and 1 -ve real root Thus, the number of real roots = 2 + 1 = 3 Hence, the number of imaginary roots = 9 - 3 = 6