Question

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

Answer 1

Give equation, $x^9-x^5+x^4+x^2+1=0$, has two changes of signs. Hence there can be a maximum of 2 positive roots.

$f(-x)=-x^9+x^5+x^4+x^2+1=0$, which has one changes of signs. Hence the given equation has a maximum of 1 negative roots.

Now, as the equation is of $9^{th}$ degree, it must have at least $(9-2-1)=6$ imaginary roots.