Question

Which of the following expressions is polynomial?

• A. $1-\sqrt{5x}$
• B. $\frac{(x-2)(x-4)}{x}$
• C. $\frac{1}{x+1}$
• D. $\frac{1}{7}{a}^3-\frac{2}{\sqrt{3}}{a}^2+4a-7$

Answer 1

The correct option is D

$\frac{1}{7}{a}^3-\frac{2}{\sqrt{3}}{a}^2+4a-7$

Explanation of correct option :

Concept: The general form of a polynomial is $f\left(x\right)=a_0+a_{1}x\mathit{+}{a}_{\mathit{2}}{x}^{\mathit{2}}\mathit{+}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{+}{a}_n{x}^n;$where $\mathit{}{a}_{\mathit{0}}\mathit{,}\mathit{}{a}_{\mathit{1}}\mathit{,}\mathit{}{a}_{\mathit{2}}\mathit{,}\mathit{}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{,}\mathit{}{a}_n\mathit{}$are real numbers and all the index of $x$ are whole numbers.

Option (D): $\frac{1}{7}{a}^3-\frac{2}{\sqrt{3}}{a}^2+4a-7$

In this expression, the exponent of $a$ in each term is a whole number,

So, this algebraic expression is a polynomial.

Explanation of incorrect options :

Option (A): $1-\sqrt{5x}=1-\sqrt{5}{x}^{\frac{1}{2}}$

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

Hence, the given algebraic expression is not a polynomial.

Option (B): $\frac{(x-2)(x-4)}{x}=\frac{x^2-6x+8}{x}=x\mathit{-}6+\frac{\mathit{8}}{x}=x\mathit{-}6+8x^{-1}$

Here, the exponent of $x$ in the third term i.e., $8x^{-1}$ is $-1$, which is not a whole number.

So, the given algebraic expression is not a polynomial.

Option (C): $\frac{1}{x+1}={(x+1)}^{-1}$

which cannot be reduced to an expression in which the exponent of the variable x have only whole numbers in each of its terms.

So, this algebraic expression is not a polynomial.

Hence the correct option is (D).