Question

Write the correct answer in the following:
Which of the following is a polynomial\[?\]
A. \[\dfrac{{{x^2}}}{2} - \dfrac{2}{{{x^2}}}\]
B. \[\sqrt {2x} - 1\]
C. \[{x^2} + \dfrac{{3{x^{\dfrac{3}{2}}}}}{{\sqrt x }}\]
D. \[\dfrac{{x - 1}}{{x + 1}}\]

Answer 1

Hint: First we have to know that the polynomials are algebraic expressions and they are expressed as a sum of the terms of the form \[a{x^n}\], where \[a\] is any real number and \[n\] is a positive integer. Hence considering each expression and reducing it in the form of \[a{x^n}\], where \[n\] must be a positive integer. If the expression is not reduced then the expression is not a polynomial.

Complete answer:
A. \[\dfrac{{{x^2}}}{2} - \dfrac{2}{{{x^2}}}\] ---(1)
The expression (1) can be rewritten as
\[\dfrac{{{x^2}}}{2} - 2{x^{ - 2}}\]---(2)
In the second term of the expression (2), the exponent of \[x\]is \[ - 2\], which is not a positive integer.
So, the expression (1) is not a polynomial.

B. \[\sqrt {2x} - 1\] ---(3)
The expression (3) can be rewritten as
\[\sqrt 2 {x^{\dfrac{1}{2}}} - 1\]---(4)
In the first term of the expression (4), the exponent of \[x\]is \[\dfrac{1}{2}\], which is not an integer.
So, the expression (3) is not a polynomial.

C. \[{x^2} + \dfrac{{3{x^{\dfrac{3}{2}}}}}{{\sqrt x }}\] ---(5)
The expression (5) can be rewritten as
\[{x^2} + 3x\] ---(6)
In the expression (6), All exponents of \[x\]are positive integers.
So, the expression (5) is a polynomial.

D. \[\dfrac{{x - 1}}{{x + 1}}\] ---(7)
The expression (7) cannot be expressed as the standard form of a polynomial.
So, the expression (7) is not a polynomial.
Hence the option (C)\[{x^2} + \dfrac{{3{x^{\dfrac{3}{2}}}}}{{\sqrt x }}\] is correct.

Therefore, the correct option is C

Note: Note that the degree of a polynomial is the highest exponent of the variable (say \[x\]) in the Polynomial. Also, a polynomial whose degree is one is called a linear polynomial. For example, \[a{x^2} + bx + c\]is a quadratic polynomial of degree \[2\] in \[x\] with real coefficients \[a,b,c\] where \[a \ne 0\].