''Reading'' polynomials at the first glance

''Reading'' polynomials at the first glance

I'm reading Proofs from the Book, and I ran into following theorem:
Suppose all roots of polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0$ are
real. Then the roots are contained in the interval:
$$ - \frac{a_{n-1}}{n} \pm \frac{n-1}{n} \sqrt{a_{n-1}^2 - \frac{2n}{n-1}
a_{n-2} } $$
So, if you know that all the roots of polynomial are real, you can get an
interval that contains them just looking at the first two coefficients.
I'm interested in other theorems/tricks that let you figure out
interesting things about a polynomial just by ''eyeing'' it. Especially if
they are surprising!