Abstract. This note is concerned with examining nilradicals and Jacob-son radicals of polynomial rings when related factor rings are Armendariz.Especially we elaborate upon a well-known structural property of Armen-dariz rings, bringing into focus the Armendariz property of factor rings byJacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) isnil when a given ring R is Armendariz, where J(A) means the Jacobsonradical of a ring A. A ring will be called feckly Armendariz if the factorring by the Jacobson radical is an Armendariz ring. It is shown that thepolynomial ring over an Armendariz ring is feckly Armendariz, in spiteof Armendariz rings being not feckly Armendariz in general. It is alsoshown that the feckly Armendariz property does not go up to polynomialrings. 1. On radicals whenfactor rings are ArmendarizThroughout this note every ring is associative with identity unless other-wise stated. For a ring R, J(R), N ∗ (R), N ∗ (R), N 0 (R) and N(R) denotethe Jacobson radical, the prime radical, the upper nilradical (i.e., sum of allnil ideals), the Wedderburn radical (i.e., the sum of all nilpotent ideals), andthe set of all nilpotent elements in R, respectively. Following [1, p. 130], asubset of R is said to be locally nilpotent if its finitely generated subringsare nilpotent. Also due to [1, p. 130], the Levitzki radical of R, written bysσ(R), means the sum of all locally nilpotent ideals of R. It is well-knownthat N