$(a)$ $\frac{x^{2}}{2}-\frac{2}{x^{2}}=\frac{x^{2}}{2}-2 x^{-2}$

Second term is $-2 x^{-2}$. Exponent of $x^{-2}$ is $-2 ,$ which is not a whole number.

So, this algebraic expression is not a polynomial.

$(b)$ $x^{2}+\frac{3 x^{\frac{2}{3}}}{\sqrt{x}}=x^{2}+3 x$

In this expression, we have only whole number as the exponent of the variable in each them.

Hence, the given algebraic expression is a polynomial.

$(c)$ $\sqrt{2 x}-1=\sqrt{2} x^{\frac{1}{2}}-1$

First term is $\sqrt{2} x^{\frac{1}{2}}$. Here, the exponent of the second term, i.e., $x^{\frac{1}{2}}$ is $\frac{1}{2},$ which is not a whole number.

So, this algebraic expression is not a polynomial.