Which of the following best defines a polynomial?

• A. An expression with variables and positive powers only
• B. An expression containing variables raised to all powers
• C. An expression with coefficients and constants
• D. An expression combining addition, subtraction, and multiplication

Answer: (A)

A polynomial is best defined as an expression that involves variables raised to non-negative integer powers, with constants (coefficients) multiplied to those variables. This means that all the exponents of the variables in the expression must be whole numbers (0, 1, 2, ...).

The defining characteristic of a polynomial is that it does not include negative exponents, fractional exponents, or any variables in the denominator. Therefore, an expression with only positive powers aligns perfectly with the characteristics of polynomials.

While the other choices mention important aspects of mathematical expressions, they either broaden the definition beyond the constraints of polynomials or include features that introduce elements not present in polynomial expressions. For example, while expressions can indeed contain operations like addition, subtraction, and multiplication, the key defining feature that distinctly identifies a polynomial is the restriction to non-negative integer exponents. Thus, the first choice succinctly captures the essential trait of a polynomial expression.