In abstract algebra, polynomials are just tuples (a_n, ... a_0) with addition (simple elementwise) and multiplication (more involved) defined. The +s and xs in the "x²+3x+1" notation are just syntax without semantic relevance; in particular x is not a variable and the substitution a ↦ P(a), a ∈ 𝗦 for any a and 𝗦 is not defined. This is because most interesting things about polynomials can be reasoned about without assuming anything about what x is.
