Counting the Number of Real Roots of A Polynomial I am interested in solving problems which involve finding the number of real roots of any polynomial.
Suppose I take a function $$f(x)=x^6+x^5+x^4+x^3+x^2+x+1$$ This does not have any real roots but I am trying to figure out if there is some analytical way that does not involve graphing to come to this conclusion.
Using Descartes' Rule of Signs,
there are zero sign changes in $f$ so by virtue of which there are no positive roots to the polynomial.
Considering $$f(-x) = x^6-x^5+x^4-x^3+x^2-x+1$$ I concluded that there are either 6 negative, 4 negative, 2 negative or zero negative roots.
So I have 4 cases to consider :


*

*0 positive roots, 6 negative roots, 0 complex roots

*0 positive roots, 4 negative roots, 2 complex roots

*0 positive roots, 2 negative roots, 4 complex roots

*0 positive roots, 0 negative roots, 6 complex roots (The correct case)


I tried differentiating $f$ but the derivative is equally bad $$f'(x) = 6x^5+5x^4+4x^3+3x^2+2x+1$$ I am unable to conclude anything from this.
I tried going about the problem the other way. If a polynomial with an even degree is always positive or negative depending on the leading coefficient, it will not have any real roots but then again, finding the extrema of the function is proving to be extremely difficult. 
I have tried using Bolzano's Intermediate Value Theorem. It guarantees the existence of at least one root but then again, there is a possibility that there might be more than one which can only be eliminated by monotonicity which again brings me back to the bad derivative.
I believe there need to be some general rules by virtue of which, we are able to calculate the number of roots for any polynomial.


*

*Is graphing the best technique for polynomials like these and if it is, are there any ways by which a quick but accurate plot can be drawn?

*While reading about the relevant theory, I came across Sturm's Method and the Newton-Raphson Method but haven't touched these yet. Is it absolutely required to know these concepts to effectively draw conclusions?

*Have I missed something?


While this method isn't guaranteed to work on all polynomials, it is surprisingly effective sometimes: notice that
\begin{align*}
[x^6+x^5+x^4+x^3+x^2+x+1]
&= x^4\left(x+\tfrac{1}{2}\right)^2+\big[\tfrac{3}{4}x^4+x^3+x^2+x+1\big] \\
&= x^4\left(x+\tfrac{1}{2}\right)^2+\tfrac34x^2\left(x+\tfrac23\right)^2+\big[\tfrac{2}{3}x^2+x+1\big] \\
&= x^4\left(x+\tfrac{1}{2}\right)^2+\tfrac34x^2\left(x+\tfrac23\right)^2+\tfrac23\left(x+\tfrac34\right)^2+\tfrac58.
\end{align*}
(Each square was chosen to eliminate the top two terms of the preceding polynomial in square brackets; this is like "completing the square" from high school math.) In this form, the polynomial is clearly always positive.
