Which one of the following is a polynomial?

(A) $\frac{x^{2}}{2}-\frac{2}{x^{2}}$
(B) $\sqrt{2 x}-1$

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

Given :

The given expressions are,

(A)  $\frac{x^{2}}{2}-\frac{2}{x^{2}}$

(B) $\sqrt{2 x}-1$

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

To do :

We have to find which of the given expressions is polynomial.

Solution :

Polynomials: Polynomials are expressions in which each term is a constant multiplied by a variable raised to a whole number power.

(A) $\frac{x^{2}}{2}-\frac{2}{x^{2}}$ is not a polynom[Math Processing Error]ial because the term $- \frac{2}{x^2}$ is equal to $-2x^{-2}$ and in this term the variable x is raised to the power $-2$ which is not a whole number.

So, $\frac{x^{2}}{2}-\frac{2}{x^{2}}$ is not a polynomial.

So, $\sqrt{2 x}-1$ is no a polynomial.

(C) $ x^{2}+\frac{3 x^{\frac{3}{2}}}{\sqrt{x}} = x^2 + 3 x^{\frac{3}{2} - \frac{1}{2}} = x^2 + 3x$. Here, the variables(x) in the terms are raised to a whole number power.

Therefore, option (C) $ x^{2}+\frac{3 x^{\frac{3}{2}}}{\sqrt{x}}$ is a polynomial. 

$which\; is\; not\; a\; whole\; number.\; So,\surd 2x\; -1\; is\; not\; a\; polynomial.$