Question

Which of the following is not a polynomial?

• A. \( x^2 + \sqrt{5} \)
• B. \( 9x^2 - 4x + \sqrt{2} \)
• C. \( \frac{1}{2} x^3 + \frac{3}{5} x^2 + 8 \)
• D. \( x + \frac{3}{x} \)

Hint:

When checking if an expression is a polynomial: - Ensure all exponents of the variable are non-negative integers. - Coefficients can be any real number, but exponents must not be negative or fractional.

Answer 1

The Correct Option is D

Step 1: Recall that a polynomial is an expression that consists of terms of the form \( ax^n \), where \( a \) is a constant and \( n \) is a non-negative integer. The exponents of \( x \) must be non-negative integers, and the coefficients can be any real number. Step 2: Analyzing the options: - (A) \( x^2 + \sqrt{5} \) is a polynomial, as the exponent of \( x \) is a non-negative integer and \( \sqrt{5} \) is a constant. - (B) \( 9x^2 - 4x + \sqrt{2} \) is a polynomial, as all exponents of \( x \) are non-negative integers. - (C) \( \frac{1}{2} x^3 + \frac{3}{5} x^2 + 8 \) is a polynomial, as the exponents of \( x \) are non-negative integers. - (D) \( x + \frac{3}{x} \) is not a polynomial, as \( \frac{3}{x} \) involves a negative exponent (\( x^{-1} \)). Thus, the correct answer is (D).