Question

Which one of the following is a polynomial?

• A. $\sqrt{3y}+5$
• B. $\frac{x^2-1}{x^2+1}$
• C. $\frac{x^2}{3}-\frac{2}{x^2}$
• D. $x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}$

Answer 1

The correct option is D

$x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}$

Step 1. Explanation for correct option:

Option D. The power of $x$ in a polynomial must be a non-negative integer.

$$\begin{gathered} x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}=x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{x^{{\displaystyle \frac{1}{2}}}} \\ \Rightarrow x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}=x^3+4x^{\frac{3}{2}}{x}^{-\frac{1}{2}} \\ \Rightarrow x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}=x^3+4{x^{\frac{3}{2}}}^{-\frac{1}{2}} \\ \Rightarrow x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}=x^3+4x^1 \\ \Rightarrow x^3+\frac{4x^{{\displaystyle \frac{3}{2}}}}{\sqrt{x}}=x^3+4x \end{gathered}$$

Since the exponent of $x$ are $3\mathrm{and}1$, which are non-negative integers, it is a polynomial.

Step 2. Explanation for incorrect options:

Option (A) $\sqrt{3y}+1=\sqrt{3}{y}^{\frac{1}{2}}+1$

Since the exponent of $y$ is $\frac{1}{2}$, that is not a whole number, it is not a polynomial.

Option (B) $\frac{x^2-1}{x^2+1}$

Since it is a rational function, it is not a polynomial.

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

Since one of the exponents of $x$ is $-2$, which is not a non-negative integer, it is not a polynomial.

Hence, the correct option is “Option D”.