It's good for your soul. There's a tradition of algebraic geometers doing things because they work. I'm teaching 241 at the same time, and we just did Stokes Theorem, so that should be clear to them. I didn't tell them that, but I was tempted. Writing the polynomial this way is the difference between algebraists and analysts.
Analysts write the coefficients in the other order because they think they're all power series.
With the usual topology on R instead of the bogus one we have been using. I mean the usual topology on R in the sense that the Math 141 students would know about if we taught them what topology was. There are logicians here. That's the problem with trying to do both geometry and number theory. All of the letters are overloaded.