Let $n\ge2$ and let $p(x)=x^n+a_{n-1}x^{n-1} \cdots a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k(<n)$ the polynomial $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}$$
Source: India Postals 2015
Tags: algebra, polynomial
Let $n\ge2$ and let $p(x)=x^n+a_{n-1}x^{n-1} \cdots a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k(<n)$ the polynomial $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}$$