To determine which of the given expressions is a polynomial, we need to understand the definition of a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers, with coefficients that are real numbers.
Let's analyze the options step by step:
### Step 1: Understand the Definition of a Polynomial
A polynomial can be expressed in the form:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \]
where:
- \( n \) is a non-negative integer (0, 1, 2, ...).
- The coefficients \( a_n, a_{n-1}, ..., a_1, a_0 \) are real numbers.
### Step 2: Analyze Each Option
Let's denote the four options as A, B, C, and D. We will check each one to see if it meets the criteria of a polynomial.
1. **Option A**: \( \frac{1}{x^{1/2}} \)
- The exponent \( 1/2 \) is not a non-negative integer. Therefore, this is **not a polynomial**.
2. **Option B**: \( \frac{x - 1}{x} \)
- This can be simplified to \( 1 - \frac{1}{x} \). The term \( \frac{1}{x} \) has a negative exponent when moved to the numerator, which means it is **not a polynomial**.
3. **Option C**: \( \frac{x^2}{x^2} \)
- This simplifies to \( 1 \), which is a constant polynomial (degree 0). Therefore, this is a **polynomial**.
4. **Option D**: \( x^2 + \frac{x^{3/2}}{x^{1/2}} \)
- The term \( \frac{x^{3/2}}{x^{1/2}} \) simplifies to \( x^{3/2 - 1/2} = x^1 \), but \( x^{3/2} \) has a non-integer exponent. Therefore, this expression is **not a polynomial**.
### Conclusion
After analyzing all options, we find that:
- **Option C** is the only expression that qualifies as a polynomial.
### Final Answer
The correct answer is **Option C**.
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