Suppose that \( p(x) \) is a degree \( n \) polynomial with complex coefficients such that \( p(x) \geq 0 \) for any real number \( x \). Prove that
\[
p(x) + p'(x) + \dots + p^{(n)}(x) \geq 0
\]
for any real number \( x \).
Prove or disprove the following:
For any positive integer \( n \), there exists a polynomial \( P_n \) of degree \( n^2 \) such that
(1) all coefficients of \( P_n \) are integers with absolute value at most \( n^2 \), and
(2) \( 1 \) is a root of \( P_n =0 \) with multiplicity at least \( n \).
For \(n\ge 1\), let \(f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k \) be a polynomial with real coefficients. Prove that if \(f(x)>0\) for all \(x\in [-2,2]\), then \(f(x)\ge 4\) for some \(x\in [-2,2]\).
Let \[p(x)=x^n+x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1 x_1 + a_0\] be a polynomial. Prove that if \(p(z)=0\) for a complex number \(z\), then \[ |z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.\]
Find all polynomials \( P(x) = a_n x^n + \cdots + a_1 x + a_0 \) satisfying (i) \( a_n \neq 0 \), (ii) \( (a_0, a_1, \cdots, a_n) \) is a permutation of \( (0, 1, \cdots, n) \), and (iii) all zeros of \( P(x) \) are rational.
Determine all polynomials \( P(z) \) with integer coefficients such that, for any complex number \( z \) with \( |z| = 1 \), \( | P(z) | \leq 2 \).
Let \( p \) be a prime number. Let \( S_p \) be the set of all positive integers \( n \) satisfying
\[
x^n – 1 = (x^p – x + 1) f(x) + p g(x)
\]
for some polynomials \( f \) and \( g \) with integer coefficients. Find all \( p \) for which \( p^p -1 \) is the minimum of \( S_p \).
Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.
Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of f´(x) is between rn-1 and rn. Prove that q>(rn-1+rn)/2.