''Reading'' polynomials at the first glance

Question

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!

Answer 1

Let $f(x) = x^n + a_{n-1}x^{n-1} + \ldots + a_1x+a_0\in \mathbb{Z}[x]$.

$\bullet$ The Perron Irreducibility Criterion states that if $a_0\neq 0$ and $|a_{n-1}|>1+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $f$ irrreducible in $\mathbb{Z}[x]$, (consequently over the rationals too, by the second Gauss's lemma).

$\bullet$ The well-known Eisenstein's criterion states that if there exists a prime number $p$ such that

1. $p$ divides $a_i$, for all $i\in \{1,\ldots, n-1\}$,
2. $p$ does not divide $a_n$,
3. $p^2$ does not divide $a_0$,

then $f$ is irreducible in $\Bbb Z[x]$ and in $\Bbb Q[x]$.

Answer 2

The Eneström–Kakeya theorem:

If $P(z)=a_nz^n+\cdots+a_0$ where $$0<a_0<a_1<\cdots<a_n,$$ then all the zeros of $P$ lie in the unit disc $|z|<1$ in the complex plane.

(See this question for a proof.)