Which of the following is a polynomial?

1. $x - \dfrac{1}{x}$

2. $x + \sqrt{x}$

3. $\dfrac{x}{\sqrt{2}} - \dfrac{1}{\sqrt[3]{2}}$

4. $x^{\dfrac{1}{2}} + x^{\dfrac{-1}{2}}$

In a polynomial, the variables must have whole number exponents (0, 1, 2…). They cannot be in the denominator or under a root.

Analysis:

1. $1/x$ is x^-1 (negative). It is not a polynomial.

2. $\sqrt{x}$ is x^1/2 (fraction). It is not a polynomial.

3. $\dfrac{x}{\sqrt{2}} - \dfrac{1}{\sqrt[3]{2}}$ has a variable with power 1. The root is only on the constant. So, it is a polynomial.

4. $x^\dfrac{1}{2} + x^\dfrac{-1}{2}$ are fractions. It is not a polynomial.

Hence, option 3 is the correct option.