To determine which of these expressions is a polynomial, we need to understand what qualifies as a polynomial expression.

A polynomial expression:

1. Consists of terms with variables raised to non-negative integer powers.
2. Does not have variables in the denominator or any fractional exponents for the variable.

Let's examine each option:

1. $2x^{\frac{1}{2}} + 6x - 10$

   • This term has $x^{\frac{1}{2}}$, which is a fractional exponent. Since a polynomial must have whole-number exponents, this is not a polynomial.

2. $\frac{2}{x^{\frac{1}{2}}} + 5x - 14$

   • Here, $x^{\frac{1}{2}}$ is in the denominator, which implies a negative fractional exponent if rewritten (as $x^{-\frac{1}{2}}$). This is not a polynomial.

3. $2x^2 + 3x - 5$

   • All terms have non-negative integer exponents (2, 1, and 0). This meets the requirements of a polynomial. This is a polynomial expression.

4. $\frac{2}{x^2} + 9x - 10$

   • The term $\frac{2}{x^2}$ places $x$ in the denominator, which translates to $x^{-2}$. Therefore, this is not a polynomial.

Answer: The correct choice is Option 3: $2x^2 + 3x - 5$.

Would you like further details on why other options are not polynomials or have any other questions?

Here are five related questions to deepen your understanding:

1. What are the key properties of polynomial functions?
2. How do negative or fractional exponents affect the classification of an expression as a polynomial?
3. Why is the variable in a polynomial required to have non-negative integer powers?
4. Can a polynomial have variables in the denominator?
5. What is the general form of a polynomial?

Tip: Remember, a polynomial's exponents must be whole numbers, and variables should not appear in denominators.