MathDB

Find all real polynomials

p(x)

of degree

n \ge 2

for which there exist real numbers

r_1 < r_2 < ... < r_n

such that (i) p(r_i) \equal{} 0, 1 \le i \le n, and (ii) p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1. Follow-up: In terms of

n

, what is the maximum value of

k

for which

k

consecutive real roots of a polynomial

p(x)

of degree

n

can have this property? (By "consecutive" I mean we order the real roots of

p(x)

and ignore the complex roots.) In particular, is k \equal{} n \minus{} 1 possible for

n \ge 3

Problem Statement