Question

Which of the following expressions is a polynomial ?

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

Answer 1

Answer: (C)

$\sqrt{2}x+x^3+3x^2$

We know that a polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power (usually positive whole number) and multiplied by a coefficient.

The simplest polynomials have one variable.

A one-variable polynomial of degree $n$ has the following form:

$a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_2{x}^2+a_1{x}^1+a$

where, the $a$'s represent the coefficients and $x$ represents the variable.

Now, consider the given options:

(a) $x+\frac{1}{x}$ can be rewritten as $x+x^{-1}$, which is not a polynomial because the exponent of the variable $x$ is $-1$ which is a negative number.

(b) $\sqrt{x}+x+x^2$ can be rewritten as $x^{\frac{1}{2}}+x+x^2$, which is not a polynomial because the exponent of the variable $x$ is $\frac{1}{2}$, which is not a whole number.

(c) $\sqrt{2}x+x^3+3x^2$ is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. Also, the exponent of the variable $x$ is a whole number.

Therefore, $\sqrt{2}x+x^3+3x^2$ is a polynomial.

(d) $x^2+x^{-2}+2$ which is not a polynomial because the exponent of the variable $x$ is $-2$, which is a negative number.

Hence, the expression $\sqrt{2}x+x^3+3x^2$ is not a polynomial.