Question 173
Given a polynomial $P(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}$ 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 $k+1$.
(Note: we call a real number $x_{0}$ a root of multiplicity $s$ of a polynomial $R(x)$ of real coefficients if there exists a polynomial $Q(x)$ such that $R(x) = (x - x_{0})^{s} Q(x)$ and $Q(x_{0}) \neq 0$.)
(Note: we call a real number $x_{0}$ a root of multiplicity $s$ of a polynomial $R(x)$ of real coefficients if there exists a polynomial $Q(x)$ such that $R(x) = (x - x_{0})^{s} Q(x)$ and $Q(x_{0}) \neq 0$.)