R, the ring of polynomials in one variable

The ring of polynomials in one variable, denoted as R[x], consists of all expressions formed by combining elements of a base set R (like numbers) with powers of a variable x, using addition and multiplication. Each polynomial is a finite sum of terms such as a₀ + a₁x + a₂x² + ..., where coefficients a₀, a₁, etc., belong to R. This structure allows for algebraic operations on polynomials while respecting their degrees and coefficients, enabling systematic study and manipulation within algebraic frameworks.