Question

Among the following expressions, the one that is a polynomial is:

• A. \( x + \frac{1}{x} \)
• B. 9
• C. \( \sqrt{x} + x^2 \)
• D. \( 5x^{-2} + 7 \)

Hint:

Polynomials can only have non-negative integer powers of \( x \), and constants are also considered polynomials.

Answer 1

The Correct Option is B

A polynomial is a mathematical expression involving a sum of powers of a variable with coefficients that are real or complex numbers. The powers must be non-negative integers (i.e., \( 0, 1, 2, 3, \ldots \)). Let’s evaluate each option: % Option (1) \( x + \frac{1}{x} \): This is \(\mathbf{not}\) a polynomial because \( \frac{1}{x} \) has a negative exponent \( x^{-1} \). A polynomial cannot have negative exponents. Hence, option (1) is not a polynomial. % Option (2) 9: This is a constant. Constants are considered polynomials of degree 0. A constant can be written as \( 9x^0 \), where the exponent is \( 0 \), which is a valid non-negative integer. Hence, option (2) is a polynomial. % Option (3) \( \sqrt{x} + x^2 \): The term \( \sqrt{x} \) can be written as \( x^{\frac{1}{2}} \), which is not a non-negative integer exponent. Since polynomials only allow non-negative integer exponents, option (3) is not a polynomial. % Option (4) \( 5x^{-2} + 7 \): The term \( 5x^{-2} \) contains a negative exponent \( x^{-2} \), which violates the condition for being a polynomial. Hence, option (4) is also not a polynomial. Thus, the correct answer is (2) 9.