When does a polynomial have a unique real root? Let $p(x)$ be an arbitrary polynomial with real coefficients. Is there a convenient way to determine whether $p(x)$ has a unique real root (not counting multiplicity)?
Edit: I know about Strum's Theorem, but I was hoping that in this special case, there would be a simpler way than computing the entire sequence of Strum polynomials. 
 A: One obviously sufficient answer is: if it derivate is always nonnegative or nonposititive, then is must have unique real root. But this is not necessary to have unique root.  
A: Here's the best I came up with:
If $p$ has odd degree, it has a real root $a$ so divide through by $x-a$ to the power of its multiplicity and continue as in the even case to verify there are $0$ real roots left.
Let $a_n$ be the leading coefficient of $p$. If $p$ has even degree, find the real roots $r_1,...r_k$ of $p'(x)$ and consider $p(r_1),...p(r_k)$. If they all have the same sign as $a_n$ there are no real roots, if one is $0$ and all others have the same sign as $a_n$ there is exactly one real root, otherwise there's more than one real root.
A: Whenever Rolle's Theorem can not hold such that the polynomial does not satisfy the condition p(a)=p(b)= 0 since p'(x) will have no real roots.
In this instance a and b are assumed to be roots of p(x) and this raises a contradiction if p'(x) has no real roots and we conclude that p(x) has a distinct root since none are found at a and b.
I know this may not be exactly what you were looking for but I hope it helps.
