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Monic Polynomials divisible by powers of (x-1)

Source: India Postals 2015

December 2, 2015
algebrapolynomial

Problem Statement

Let n≥2n\ge2 and let p(x)=xn+an−1xn−1⋯a1x+a0p(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)k(<n) the polynomial (x−1)k+1(x-1)^{k+1} divides p(x)p(x) then ∑i=0n−1∣ai∣≥1+2k2n\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}
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