To determine which of the given expressions is a polynomial in one variable, we need to understand the definition of a polynomial. A polynomial in one variable \( x \) is an expression of the form:
\[
a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0
\]
where:
- \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients (which can be any real numbers),
- \( n \) is a non-negative integer (0, 1, 2, ...),
- The powers of \( x \) must be whole numbers (non-negative integers).
Now, let's analyze the options given in the question one by one to see which one meets the criteria of a polynomial in one variable.
### Step 1: Analyze the first option
- **Expression:** \( \frac{1}{x} \)
- **Analysis:** The variable \( x \) is in the denominator, which means it has a negative exponent of -1. This does not satisfy the condition of having non-negative integer powers.
- **Conclusion:** This is **not** a polynomial.
### Step 2: Analyze the second option
- **Expression:** \( x^{1/3} \)
- **Analysis:** The exponent \( 1/3 \) is a rational number, not a non-negative integer. This does not satisfy the polynomial condition.
- **Conclusion:** This is **not** a polynomial.
### Step 3: Analyze the third option
- **Expression:** \( 2x^2 + 3x + 1 \)
- **Analysis:** The exponents are 2, 1, and 0, which are all non-negative integers. The coefficients (2, 3, and 1) are real numbers.
- **Conclusion:** This **is** a polynomial.
### Step 4: Analyze the fourth option
- **Expression:** \( 4x^2 + 5y + 6z \)
- **Analysis:** This expression contains three different variables: \( x, y, z \). A polynomial in one variable must only contain one variable. Therefore, this does not meet the criteria.
- **Conclusion:** This is **not** a polynomial in one variable.
### Final Conclusion
Based on the analysis, the only expression that qualifies as a polynomial in one variable is:
**Answer:** Option C: \( 2x^2 + 3x + 1 \)
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