Using Rouché prove that $a_0\geq a_1 \geq ... \geq a_n > 0$ the polynomial $P(z)=a_n z^n + ... + a_1 z + a_0$ has no roots $|z|<1$
135385
$0 < a_0 \leq a_1 \leq \cdots \leq a_n$, show $ p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0$ has no root in $|z| < 1$
109102
Determine all real polynomials with $P(0)=0$ and $P(x^2+1)=(P(x))^2+1$
40014
A polynomial that is zero on an open set
57191
Show that if $a>e$, the equation $az^n=e^z$ admit $n$ roots in the unit disk - Rouché theorem
89381
Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots a_n$ cannot be completely resolved into linear factors if $a_1^2<a_2$.
76696
Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
115583
prove that $a^2(p-q)(p-r)+b^2(q-p)(q-r)+c^2(r-p)(r-q)\ge 0$
127547
Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
153107
Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
5410
Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
66707
How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $
40502
The Degree of Zero Polynomial.
91388
Prove that $f$ is a polynomial if one of the coefficients in its Taylor expansion is 0
4706
Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
82204
Prove $ \lim_{n\to\infty}\sqrt[n]{p(n)}=1 $ for polynomial
53669
Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$