Can't you just define polynomials as finite sequences of rationals with nonzero last term? E.g. encode x^2 + 2x -1/2 as (-1/2, 2, 1), and define evaluation at a given point in the obvious way.
You generally define them as infinite sequences with a finite number of non-zero terms. Since we only care about a specific quotient field you could force it to work by working with only polynomials of bounded degree (<=2 should work. <=4 is probably the least bad option), but it would get ugly.
You could also ignore polynomials entirely. Since Q(sqrt(2)) is just a two dimensional vector space over Q, we could define it as the ordered pairs Q^2 with an appropriate definition of multiplication and division (like we often define the complex numbers to highshoolers), but this also gets ugly.
I guess the moral of this post is 99% of the time a finitist or constroctivist complains you can rework your theory into a more ugly one that avoids the complaint.
