Which of the following expressions is a polynomial in $x$?

• A. $x^{\frac{1}{2}}-4$
• B. $2/x-3$
• C. $\frac{5}{x}+1$
• D. $3x^4-2x+7$

Answer: (D)

Verified step by step guidance

1. Recall that a polynomial in \(x\) is an expression consisting of terms with non-negative integer exponents of \(x\), combined using addition, subtraction, and multiplication by constants.
2. Examine the first expression: \(x^{\frac{1}{2}} - 4\). Since the exponent \(\frac{1}{2}\) is not an integer, this is not a polynomial.
3. Look at the second expression: \(\frac{2}{x - 3}\). This is a rational expression with \(x\) in the denominator, which disqualifies it from being a polynomial.
4. Consider the third expression: \(\frac{5}{x} + 1\). Here, \(x\) is in the denominator, so this is not a polynomial either.
5. Finally, analyze the fourth expression: \$3x^{4} - 2x + 7\(. All exponents of \)x\( are whole numbers (4 and 1), and the expression is a sum of terms with constants and powers of \)x$, so this is a polynomial.