Z[x], the ring of polynomials with integer coefficients

Z[x] is the set of all polynomials where the coefficients are integers (whole numbers, positive, negative, or zero), and the variable is x. Think of each polynomial as a mathematical expression like 3x² - 5x + 7. Addition and multiplication of these polynomials follow the usual rules, and the coefficients always remain integers. This structure allows mathematicians to study properties of polynomials with integer coefficients systematically, similar to how integers form a familiar system of numbers, but extended to include powers of a variable.