Number of real zeros of a polynomial Given a $n$-th degree polynomial with real coefficients 
$$c_nx^n + c_{n-1}x^{n-1} + ... + c_1x+c_0$$
Is there any theorem that tells me how many zeros are real (zero imaginary part). Anything related would help. Thanks!
 A: Sturm's theorem might help!
Sturm's Theorem: If $f(x)$, freed from equal roots, be divided by $f'(x)$, and the last divisor by the last remainder, changing the sign of each remainder before dividing by it, until a remainder independent of $x$ is obtained, or else a remainder which cannot change its sign; then $f(x)$, $f'(x)$, and the successive remainders constitute Sturm's functions, and are denoted by $f(x),f_1(x),f_2(x),\,\&\text{c}\ldots\ldots f_m(x)$.
The operation may be exhibited as follows:$$\begin{align*} & f(x)=q_1f_1(x)-f_2(x),\\ & f_1(x)=q_2f_2(x)-f_3(x),\\ & f_2(x)=q_3f_3(x)-f_4(x),\\ & \ldots\hspace{7mm}\ldots\hspace{7mm}\ldots\hspace{7mm}\ldots\\ & f_{m-2}(x)=q_{m-1}f_{m-1}(x)-f_m(x).\end{align*}$$
Note: Any constant factor of a remainder may be rejected, and the quotient may be set down for the corresponding function.


Example:

To find the position of the roots of$$f(x):=x^4-4x^3+x^2+6x+2=0$$
Sturm's functions, formed as the rule above, are calculated as$$\begin{align*} & f(x)=x^4-4x^3+x^2+6x+2\\ & f_1(x)=2x^3-6x^2+x+3\\ & f_2(x)=5x^2-10x-7\\ & f_3(x)=x-1\\ & f_4(x)=12\end{align*}$$
The first terms of the functions are all positive, therefore there is no imaginary root.
There are changes of signs in the functions as $x$ passes through a specific integral.
$$\begin{array}{|r|c|c|c|c|c|c|c|}\hline\\x= & -2 & -1 & 0 & 1 & 2 & 3 & 4\\\hline\\f(x)= & + & + & + & + & + & + & +\\f_1(x)= & - & - & + & + & - & + & +\\f_2(x)= & + & + & - & - & - & + & +\\f_3(x)= & - & - & - & + & + & + & +\\f_4(x)= & + & + & + & + & + & + & +\\\hline\\\text{No. of changes} & 4 & 4 & 2 & 2 & 2 & 0 & 0\\\text{of sign}\small\ldots\ldots\\\hline\end{array}$$
There are two changes of sign lost while $x$ passes from $-1$ to $0$, and two more while $x$ passes from $2$ to $3$. Therefore, there are two roots lying between $0$ and $-1$; and two roots also between $2$ and $3$. These roots are also incommensurable.
