Finding real roots of a Polynomial Equation without graphs. I am interested in finding the number of real roots of the polynomial equation
$$
x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} = 0.
$$ 
I know that graphing it would tell me how many real roots it has: the graph cuts the x-axis three times. But the coefficients are telling me some factorization is possible. I tried to write it like
$$
\left(x^3 + \frac{3}{8}\right)^3 = x - \frac{3}{8},
$$
but what next? Or is a graphical solution is the only possibility?

We have
$$
\begin{align}
&x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} \\
&\qquad = \frac{1}{512} (2 x-1) \left(4 x^2+2 x-3\right) \left(64 x^6+64 x^4+48 x^3+64 x^2+24 x+73\right).
\end{align}
$$
Since
$$
\begin{align}
&64 x^6+64 x^4+48 x^3+64 x^2+24 x+73 \\
&\qquad \geq 64 x^6+48 x^3+24 x+73 \\
&\qquad > 28 x^6+48 x^3+24 x+44 \\
&\qquad = 4 (x+1)^2 \left(7 x^4-14 x^3+21 x^2-16 x+11\right) \\
&\qquad \geq 4 (x+1)^2 \left(7 x^4-14 x^3+15 x^2-16 x+8\right) \\
&\qquad = 4 (x+1)^2 (x-1)^2 \left(7 x^2+8\right) \\
&\qquad \geq 0,
\end{align}
$$
the only real roots are $x=1/2$ and
$$
x = \frac{-1 \pm \sqrt{13}}{4}.
$$
