Final answer:
Options A, B, and C are polynomials as their terms have non-negative integer exponents and no variables in the denominator, provided option C is interpreted correctly, either as ¹/₃x² or (2/3)x². Options D and E are not polynomials due to variables in the denominator.
Step-by-step explanation:
A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An equation is a polynomial if all the exponents of the variables are whole numbers and there are no variables in the denominator.
Considering the given options:
- A. (x²/3 + 0x + 1) is a polynomial because all terms have non-negative integer exponents and no variables in the denominator.
- B. (x³ + 2x + 2) is a polynomial because all terms have non-negative integer exponents and no variables in the denominator.
- C. (2/3x² + x + 1) is a polynomial assuming that it is written as (¹/₃x² + x + 1), with all terms having non-negative integer exponents and no variables in the denominator. If it's (2/3)x², it is still a polynomial as the coefficient is a constant.
- D. (2/x³ + x + 1/2) is not a polynomial because the first term has a variable in the denominator.
- E. (x² + x + 1/x² + 1) is not a polynomial because the term 1/x² has a variable in the denominator.
Therefore, options A, B, and C are polynomials, assuming C is written correctly as mentioned.