$P(x)=x^n+a_1x^{n−1}+\dots+a_n$ is a polynomial with real coefficients. If $a_1^2<a_2$ Prove that all roots of P(x) cannot be real.
89381
Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots a_n$ cannot be completely resolved into linear factors if $a_1^2<a_2$.
33011
Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
82771
Proving that any polynomial $p:\mathbb{R}\rightarrow \mathbb{R}$ is not bounded, but its restriction to any bounded set is bounded
87331
Real polynomial in two variables
99000
No polynomial with integer coefficients satisfies the conditions
154507
If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts
133101
Number of solutions of $P(x)=e^{ax}$ if $P$ is a polynomial
139930
Find all polynomials $p$ such that $p(x^2)=p(x)p(x+1)$
116686
$p=2^n+1$. Prove that every quadratic nonresidue modulo $p$ is a primitive root modulo $p$
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.
50084
Probability that a quadratic polynomial with random coefficients has real roots
88466
Show that polynomial $X^{n+1}-aX^n+aX-1$ has only roots of module $1$
1962
Polynomial $P(a)=b,P(b)=c,P(c)=a$
41486
If $x^n + a_{n-1}x^{n-1} + ... + a_0 = 0$ has a rational root, then such root is an integer
106011
Proving it has all real roots
143466
Roots of functions / polynomials
82567
Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $