KöMaL 2013 A580

Show that for every positive integer $k$ and real number $\varepsilon$>0 there exists a polynomial $p$($x$)=$x^n$+$a_{n-1} x^{n-1}$+...+$a_1 x$+$a_0$ with real coefficients such that $p$($x$) is divisible by the polynomial ($x$-1)$^{k+1}$ and $\sum_{\ell=0}^{n-1} |a_\ell| < 1+\frac{(2+\varepsilon)k^2}{n}.$