Problem

Source: 2016 Saudi Arabia BMO TST , level 4+, I p1

Tags: algebra, polynomial, Real Roots



Given a polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + ...+ a_1x + a_0$ of real coefficients. Suppose that $P(x)$ has $n$ real roots (not necessarily distinct), and there exists a positive integer $k$ such that $a_k = a_{k-1} = 0$. Prove that $P(x)$ has a real root of multiplicity $k + 1$.