What is the number of real roots of the polynomial The number of distinct real roots of the equation 
$x^9+x^7+x^5+x^3+x+1=0.$

Let $f(x)=x^9+x^7+x^5+x^3+x+1$, then $f^\prime (x)=9x^8+7x^6+5x^4+3x^2+1\geq1>0.$ Hence $f$ is strictly increasing, so has atmost one real root. Also $f(0)=1>0$ and $f(-1)=-4<0$ and so has a root between $-1$ and $0$.
