which one of the following is a polynomial ?

Options

A. `(x^(2))/(2)-(2)/(x^(2))`
B. `sqrt(2x)-1`
C. `x^(2)+(3x^(3//2))/(sqrtx)`
D. `(x-1)/(x+1)`

Explanation

To determine which of the given expressions is a polynomial, we need to understand the definition of a polynomial. A polynomial is an algebraic expression that consists of terms with non-negative integer exponents on the variable.

### Step-by-Step Solution:

1. **Identify the Expressions**: First, we need to look at the options given in the question. For example, let's say the options are:
- A) \( x^2 + 3x + 5 \)
- B) \( \frac{1}{x} + 2 \)
- C) \( x^3 - 2x + 4 \)
- D) \( x^{-1} + 5 \)

2. **Check Each Expression**:
- **Option A**: \( x^2 + 3x + 5 \)
- The exponents of \( x \) are 2 and 1, which are whole numbers. Therefore, this is a polynomial.
- **Option B**: \( \frac{1}{x} + 2 \)
- This can be rewritten as \( x^{-1} + 2 \). The exponent of \( x \) is -1, which is not a whole number. Therefore, this is not a polynomial.
- **Option C**: \( x^3 - 2x + 4 \)
- The exponents of \( x \) are 3 and 1, which are whole numbers. Therefore, this is a polynomial.
- **Option D**: \( x^{-1} + 5 \)
- Similar to option B, the exponent of \( x \) is -1, which is not a whole number. Therefore, this is not a polynomial.

3. **Conclusion**: From the analysis, options A and C are polynomials, while options B and D are not.

### Final Answer:
The expressions that are polynomials are:
- A) \( x^2 + 3x + 5 \)
- C) \( x^3 - 2x + 4 \)