LetF:ℝ×ℝ→ℝbe a real-valued polynomial function of the formF(x,y)=as(x)ys+as-1(x)ys-1+⋯+a0(x), where the degreesofyinF(x,y)is greater than or equal to1. For arbitrary polynomial functionf(x)∈ℝ[x],x∈ℝ, we will find a polynomial solutiony(x)∈ℝ[x]to satisfy the following equation: (*):F(x,y(x))=af(x), wherea∈ℝis a constant depending on the solutiony(x), namely, a quasi-coincidence (point) solution of (*), andais called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficientas(x)must be a factor off(x), and (ii) each solution of (*) is of the formy(x)=-as-1(x)/sas(x)+λp(x), whereλis arbitrary andp(x)=c(f(x)/as(x))1/sis also a factor off(x), for some constantc∈ℝ, provided the equation(*)has infinitely many quasi-coincidence (point) solutions.