Z[x] (polynomials with integer coefficients)

\(\mathbb{Z}[x]\) represents the set of all polynomials with integer coefficients, where the coefficients are whole numbers like -3, 0, or 5, and \(x\) is an indeterminate symbol representing a variable. Think of these polynomials as algebraic expressions involving \(x\), such as \(2x^3 - 5x + 7\). This set includes infinitely many possibilities, and the core property is that when you add or multiply any two such polynomials, the results stay within the same set, maintaining integer coefficients. It’s a fundamental structure in algebra, bridging number theory and polynomial algebra.