Which one of the following is a polynomial?
(a) $\frac{x^{2}}{2}-\frac{2}{x^{2}}$
(b) $\sqrt{2 x}-1$
(c) $x^{2}+\frac{3 x^{3 / 2}}{\sqrt{x}}$
(d) $\frac{x-1}{x+1}$
(c)
(a) Now, $\frac{x^{2}}{2}-\frac{2}{x^{2}}=\frac{x^{2}}{2}-2 x^{-2}$, it is not a polynomial, because exponent of $x$ is $-2$ which is not a whole number.
(b) Now, $\sqrt{2 x}-1=\sqrt{2} x^{1 / 2}-1$, it is not a polynomial, because exponent of $x$ is $-\frac{1}{2}$ which is not a whole number.
(c) Now, $x^{2}+\frac{3 x^{\frac{3}{2}}}{\sqrt{x}}=x^{2}+3 x^{\frac{3}{2}}-\frac{1}{2}=x^{2}+3 x^{\frac{2}{2}}=x^{2}+3 x$, it is a polynomial, because exponent of $x$ is a whole number.
(d) $\frac{x-1}{x+1}$, it is not a polynomial because it is a rational function.
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