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Which of the following are polynomials?

A. (x²/3 + 0x + 1)
B. (x³ + 2x + 2)
C. (2/3x² + x + 1)
D. (2/x³ + x + 1/2)
E. (x² + x + 1/x² + 1)

User Michaeak
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1 Answer

3 votes

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.

User Keith Holloway
by
8.2k points
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