Problem #PRU-64412

Problems Calculus Derivative Derivative and multiple roots

Problem

Prove that the polynomial $P (x) = a_{0} + a_{1} x + \dots + a_{n} x^{n}$ has a number $- 1$ which is a root of multiplicity $m + 1$ if and only if the following conditions are satisfied: $\begin{aligned} a_{0} - a_{1} + a_{2} - a_{3} + \dots + (- 1)^{n} a_{n} & = 0, \\ - a_{1} + 2 a_{2} - 3 a_{3} + \dots + (- 1)^{n} n a_{n} & = 0, \\ \dots \\ - a_{1} + 2^{m} a_{2} - 3^{m} a_{3} + \dots + (- 1)^{n} n^{m} a_{n} & = 0. \end{aligned}$

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