Let S={p1,âŠ,ps} be a finite non-empty set of distinct prime numbers, let fâZ[X] be a polynomial of degree nâ„1, and let SâČâS be the subset of all pâS such that f has a root in Zp. For any non-zero integer y, write y=p1k1âŠpsksy0, where k1,âŠ,ks are non-negative integers and y0 is an integer coprime to p1,âŠ,ps. We define the f-normalized S-part of y by [y]f,S:=p1k1rp1,S(f)âŠpsksrps,S(f), with rp,S(f)=1 if pâSâSâČ and rp,S(f)=RSâČ(f)/Rp(f) if pâSâČ, where Rp(f) denotes the largest multiplicity of a root of f in Zp and RSâČ(f):=maxpâSâČâĄRp(f). For positive real numbers Δ,B with Δ<RSâČ(f)/n, we consider the number NË(f,S,Δ,B) of integers x such that |x|â€B and 0<|f(x)|Δâ€[f(x)]f,S. We prove that if sâČ:=#SâČâ„1, then NË(f,S,Δ,B)âf,S,ΔB1â(nΔ)/RSâČ(f)(logâĄB)sâČâ1 as Bââ. Moreover, if f has no multiple roots in Zp for any pâSâČ and sâČ:=#SâČâ„2, then there exists a constant C(f,S,Δ)>0 such that NË(f,S,Δ,B)âŒC(f,S,Δ)B1ânΔ(logâĄB)sâČâ1 as Bââ.