Which algebraic expression is a polynomial in $x$?

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

Answer: (C)

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 each given expression to check the exponents of \( x \):
3. 1. \( 3 + \frac{1}{x} \) has a term \( \frac{1}{x} = x^{-1} \), which is a negative exponent, so it is not a polynomial.
4. 2. \( 2x^{\frac{1}{2}} - 1 \) has an exponent of \( \frac{1}{2} \), which is not an integer, so it is not a polynomial.
5. 3. \( 4x^{3} - 2x + 7 \) has exponents 3, 1, and 0 (constant term), all non-negative integers, so this is a polynomial.