Regarding one of Korkin-Zolotarev problems

Korkin and Zolotarev establish that the required polynomials have n roots in the interval (‒1, +1) and derive a series of equations that these roots satisfy; from the analysis of these equations they deduce the uniqueness of the solution to the problem. This note offers a simpler proof of the uniqueness of the solution, requiring knowledge only of the number of roots of the required polynomials in the interval (‒1, +1). The method of evidence can be successfully used in many similar issues of a more general nature.