That’s fine as a pure mathematics perspective, but in physics dimensions come into play often, and one often has polynomials whose domain and coefficients come “tagged” with particular dimensions.
OP did say, “speaking about math...”. The operations on polynomials and dealing with polynomials doesn’t have anything to do with units of measurement. It may be the case that when used in some areas and in some contexts that the units matter but I think it clouds issues to bring them up.
The intuition for operating with polynomials is best obtained by not worrying about units.
As he said, x is part of a ring. In physics the multiplication of two elements from the same ring end up on another ring (meters^2 for example) in which, for example, you cannot do addition with elements from the original ring (meters)
The mathematical definition of a ring precludes this from happening. By definition, in a ring when you multiply 2 elements from the ring you get another element of the ring. Perhaps you are referring tensor products (geometric algebra for physicists).
