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 evaluate each expression based on the definition of a polynomial. A polynomial is an algebraic expression that consists of terms with non-negative integer exponents.

Let's analyze each option step by step:

### Step 1: Analyze the first expression: \( \frac{x^2}{2} - \frac{2}{x} \)

- Rewrite \( \frac{2}{x} \) as \( 2x^{-1} \).
- The term \( x^{-1} \) has a negative exponent, which is not allowed in polynomials.

**Conclusion**: This expression is **not a polynomial**.

### Step 2: Analyze the second expression: \( \sqrt{2}x - 1 \)

- Rewrite \( \sqrt{2}x \) as \( \sqrt{2}x^{1/2} \).
- The exponent \( 1/2 \) is a fraction, which is not a non-negative integer.

**Conclusion**: This expression is **not a polynomial**.

### Step 3: Analyze the third expression: \( \frac{x^2 + 3x^{3/2}}{\sqrt{x}} \)

- Rewrite \( \sqrt{x} \) as \( x^{1/2} \).
- The expression becomes \( \frac{x^2 + 3x^{3/2}}{x^{1/2}} \).
- Simplifying gives us \( x^{2 - 1/2} + 3x^{3/2 - 1/2} = x^{3/2} + 3x^{1} \).
- Both \( x^{3/2} \) and \( x^{1} \) contain a term with a fractional exponent.

**Conclusion**: This expression is **not a polynomial**.

### Step 4: Analyze the fourth expression: \( \frac{x - 1}{x + 1} \)

- This is a rational expression (a fraction).
- Polynomials cannot have variables in the denominator.

**Conclusion**: This expression is **not a polynomial**.

### Final Conclusion

After analyzing all four expressions, none of them qualify as a polynomial. Therefore, the correct answer is that **none of the given expressions is a polynomial**.