Your question is not specific to polynomials, it also occurs for integers. In particular, every programming language has to deal with this question. For instance, just compare the following results (in Python, but you could choose C, Java, C++, etc.). Here % means mod.
7 + 7 % 5 9 (7 + 7) % 5 4 7 + (7 % 5) 9
You conclude that 7 + 7 % 5 was interpreted as 7 + (7 % 5). The reason is that $C$-style languages use precedence levels to avoid ambiguity. In particular, multiplication, division and modulo have higher priority than addition.
That being said, Rule 8 of The Elements of Programming Style states
Parenthesize to avoid ambiguity
which confirms Zhen Li's and Bill Dubuque's comments to your question.