Question

Which of the following is not a polynomial?

• A. \(x^2 - 7\)
• B. \(2x^2 + 7x + 6\)
• C. \(\frac{1}{2}x^2 + \frac{1}{2}x + 4\)
• D. \(x + \frac{4}{x}\)

Hint:

A quick way to spot a non-polynomial is to look for variables in the denominator of a fraction or under a radical sign (e.g., \(\sqrt{x}\) which is \(x^{1/2}\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A polynomial is an algebraic expression in which the variables involved have only non-negative integer powers. The coefficients can be any real number.

Step 2: Detailed Explanation:
Let's analyze each option:
(A) \(x^2 - 7\): The powers of x are 2 and 0 (since \(7 = 7x^0\)). Both are non-negative integers. So, this is a polynomial.
(B) \(2x^2 + 7x + 6\): The powers of x are 2, 1, and 0. All are non-negative integers. So, this is a polynomial.
(C) \(\frac{1}{2}x^2 + \frac{1}{2}x + 4\): The powers of x are 2, 1, and 0. All are non-negative integers. The coefficients (\(\frac{1}{2}, \frac{1}{2}, 4\)) are real numbers. So, this is a polynomial.
(D) \(x + \frac{4}{x}\): This expression can be rewritten using exponents as \(x^1 + 4x^{-1}\). The power of x in the second term is -1, which is a negative integer. According to the definition of a polynomial, the powers of the variable must be non-negative. Therefore, this expression is not a polynomial.

Step 3: Final Answer:
The expression \(x + \frac{4}{x}\) is not a polynomial because it contains a term with a negative exponent. This matches option (D).