Jacques Sturm developed a method which computes the number of distinct real roots of a real polynomial. This technique can be used to count the total number (of distinct) real roots or just count the number of roots lying in a particular interval.
Sturm says to take a polynomial $r_0=p(x)$ and compute its derivative $r_1=p'(x)$. Then divide $p(x)$ by $p'(x)$. This yields a polynomial quotient and remainder. Let $r_2 = - \mathrm{Remainder}(r_0,r_1)$ be the negative of the remainder. Now continue dividing $r_{i-1}$ by $r_{i}$ setting $r_{i+1}$ equal to the negative remainder until a zero remainder is achieved (this must happen eventually since each division yields a negative remainder of lower degree).
Next, take the list negative remainders (as well as $p(x)$ and $p'(x)$) and evaluate each at $x=a$. This yields a list of real numbers. Count the number of sign changes in this list (ignoring zeros) and call the number of sign changes $A$. Then do the same for $x=b$ and call the number of sign changes $B$. If neither $x=a$ nor $x=b$ is a repeated root, the number of distinct roots in the interval $I=(a,b] = \{ x\in \mathbb{R} \;|\; a \lt x\leq b\}$ is exactly $A-B$.
When either $x=a$ or $x=b$ is a repeated root, substituting that number into the sequence of polynomials will yield a string of 0's. But all is not lost, if the entire sequence is divided by its last nonzero term, the modified Sturm sequence then applies as a method.
Note that we can "evaluate the Sturm Sequence at $\pm\infty$" (where limits are actually being taken to $\pm\infty$) and use this method to find the total number of distinct real roots of $p(x)$ on the whole real line $\mathbb{R}=(-\infty,\infty)$.
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