Question

Which one of the following is a polynomial?
A. $\sqrt {3y} + 5$
B. $\dfrac{{{x^2} - 1}}{{{x^2} + 1}}$
C. $\dfrac{{{x^2}}}{3} - \dfrac{2}{{{x^2}}}$
D. ${x^3} + \dfrac{{4{x^{\dfrac{3}{2}}}}}{{\sqrt x }}$

Answer 1

Hint: In this question we will find the solution by using the basic definition of the polynomial. In a polynomial, the power of the variable involved must not be negative, that is it has to be a non-negative integer. So, we will simplify every equation given in question and verify them to find the polynomial.

Complete step by step answer:
Polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division. General polynomial of second degree is
$a{x^2} + bx + c$

Now, we will verify every equation given in question.
Option A. $\sqrt {3y} + 5$
We can write this equation as,
$\sqrt 3 {y^{\dfrac{1}{2}}} + 5$
In this equation the power of $x$ is not an integer. So, this is not a polynomial.

Now we will check the next option.
Option B. $\dfrac{{{x^2} - 1}}{{{x^2} + 1}}$
Variability of a polynomial can never be in the denominator. But in this equation the variable is in the denominator. So, it is not polynomial.

Option C. $\dfrac{{{x^2}}}{3} - \dfrac{2}{{{x^2}}}$
We can write this equation as:
$\dfrac{{{x^2}}}{3} - 2{x^{ - 2}}$
The power of variables is negative. So, this is not a polynomial.

Option D. ${x^3} + \dfrac{{4{x^{\dfrac{3}{2}}}}}{{\sqrt x }}$
We can write this equation as:
${x^3} + 4{x^{\dfrac{3}{2}}} \times {x^{ - \dfrac{1}{2}}}$
Simplifying the equation
${x^3} + 4{x^{\dfrac{3}{2} - \dfrac{1}{2}}}$
$\Rightarrow {x^3} + 4{x^{\dfrac{2}{2}}}$
$\therefore {x^3} + 4x$
The power of variables is not negative and is also an integer. So, this is a polynomial.

Hence, ${x^3} + \dfrac{{4{x^{\dfrac{3}{2}}}}}{{\sqrt x }}$ is a polynomial. So, option (D) is the correct answer.

Note: Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). Based on the numbers of terms present in the expression, it is classified as monomial, binomial, and trinomial.