Many mathematicians believe that Köthe conjecture is one of the hardest problem in mathematics. It was started in 1930 and it is still open until now. Some reformulations of the Köthe conjecture were found by many prominent authors. The Köthe conjecture is also equivalent to the condition, for any ring \(R\), the Jacobson radical of \(R[x]\) consists of the polynomials with coefficients from the upper nilradical of \(R\). Hence, research related to nilradical is interesting. On the other hand, \textit{A. Smoktunowicz} [J. Algebra 233, No. 2, 427--436 (2000; Zbl 0969.16006)] introduced a ring \(R\) such that Nil\((R[x])\) is a proper subset of Nil\((R)[x]\) where Nil\((R)[x]\) is the set of nilpotent elements in \(R[x]\) which is precisely the nilradical of \(R[x]\). In this paper, the authors provide the converse condition by constructing a ring \(R\) such that Nil\((R)^2=0\) and a polynomial \(f \in R[x] \setminus\mathrm{Nil}(R)[x]\) satisfying \(f^2=0\). However, the smallest possible degree a such polynomial is seven. This example also explains the answer of an open problem asked by \textit{R. Antoine} [Commun. Algebra 38, No. 11, 4130--4143 (2010; Zbl 1218.16013)] related to Armendariz ring.