After going through the basics of Arithmetic it isn’t surprising that the same principles that apply there also apply to other areas of mathematics. Section 4.2 is nearly identical to section 1.2. All of these principles are applicable to rings of polynomials.
So far Hungerford has looked at polynomials and x as a place holder rather than a variable. I imagine that this is the case to maintain the abstract nature of analyzing polynomials so that when returning to view them as a variable there shall be a greater understanding of how the polynomials work.