Although polynomials and polynomial functions are different mathematical concepts, they are indiscriminately used in the national curriculum. Because of this, it is suggested that students do not fully understand the concepts of variables and the didactical meaning of these expressions. In this paper, therefore, we classify expressions which appear in the secondary curriculum into numerical expressions and letter expressions, letter equations for generalized expressions and algebraic expressions, and algebraic expressions into polynomials, equations, and functional expressions. According to the classification, we define variables, and then reveal the meanings of indeterminates and variables in the polynomials and the polynomial functions respectively through theoretical study; and then study their related concepts. Equalities and the related operations are basic concepts in the study of the structure of the expressions. Related concepts to polynomial expressions are factorization, the remainder theorem, the method of undetermined coefficients, the introduction of complex numbers, and so on. On the other hand, polynomials function as concepts which deal with change and function as a basic element. Related concepts include: increasing and decreasing, maximum and minimum, point of inflection, and smooth curves and surfaces. Finally, we enhance the justification of the content in the curriculum through an analysis of the related content of text books. It is hoped that this study will promote the professional development of teachers of mathematics, will be helpful in the critical analysis of the curriculum, and will provide a foundation for Curriculum Instruction.