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

• A. $x^{-3}+2$
• B. $-13$
• C. $x^{\frac{1}{2}}-4$
• D. $\frac{5}{x}+1$

Answer: (B)

Verified step by step guidance

1. Recall that a polynomial in \( x \) is an expression consisting of terms with non-negative integer exponents of \( x \) and coefficients that are real numbers. The exponents must be whole numbers (0, 1, 2, 3, ...).
2. Examine the first expression: \( x^{-3} + 2 \). Here, the exponent of \( x \) is \( -3 \), which is a negative integer. Since polynomials cannot have negative exponents, this is not a polynomial.
3. Look at the second expression: \( -13 \). This is a constant term, which can be considered a polynomial of degree 0 because it can be written as \( -13x^0 \). Constants are always polynomials.
4. Check the third expression: \( x^{\frac{1}{2}} - 4 \). The exponent \( \frac{1}{2} \) is a fraction, not an integer, so this is not a polynomial.
5. Finally, consider the fourth expression: \( \frac{5}{x} + 1 \). This can be rewritten as \( 5x^{-1} + 1 \), which has a negative exponent, so it is not a polynomial.