Hey, thanks for asking this question. I also thought this problem was weird and stuck on it.
From reading various replies across the interwebs, my understanding is that:
- If p(x) had ended up being equal to 0, then it would have been the zero polynomial and have no degree or a degree of -1 or -infinity, depending on the math system. In all cases, this would make the statement that "p(x) is a real polynomial of degree ≤ 4" incorrect.
- If p(x) had ended up being equal to a non-zero constant, then it would not have been the zero polynomial and have a degree of 0. This would make the statement that "p(x) is a real polynomial of degree ≤ 4" correct.
Since p(x) did end up being equal to a non-zero constant 5, then it is a constant polynomial of degree 0 and so makes sense in the context of the question. A real, constant polynomial has an infinite number of distinct roots. For example, when x = -n, ..., -1, 0, 1, ..., n, p(x) = 5.
However, the question is still quite strange as it was not covered in the text and other replies stated it usually appears in formal algebraic geometry
and theories of degrees
topics that are not seen on the GRE subject test.