# Number of distinct real roots of $x^9 + x^7 + x^5 + x^3 + x + 1$

The number of distinct real roots of this equation $$x^9 + x^7 + x^5 + x^3 + x + 1 =0$$ is

Descarte rule of signs doesnt seems to work here as answer is not consistent . in general i would like to know nhow to find number of real roots of any degree equation

Your polynomial has only odd exponents (apart from the constant term, which causes a shift up the "y-axis"). So in negative x direction it is monotonely decreasing and in positive direction monotonely increasing. All in all you get exactly one real root (because at $x=0$ it has positive value and at $x=-1$ it has negative value.
Denote the polynomial as $p$.
$p$ have at least one real root because it is of odd degree (this follows from IVT).
Assume $p$ have more then one real root, then there are two different points $x_1,x_2$ s.t $p(x_1)=p(x_2)=0$
By Rolle's theorem this imply there is $x_1<c<x_2$ s.t $p'(x)=0$ - show such a point does not exist