In a polynomial, the exponents of the variables are always

Options

A. integers
B. positive integers
C. non-negative integers
D. non-positive integers

Answer

To solve the question "In a polynomial, the exponents of the variables are always...", we need to understand the definition and properties of polynomials.

### Step-by-Step Solution:

1. **Definition of a Polynomial**: A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers and coefficients.

2. **Understanding Exponents**: The exponents of the variables in a polynomial must be whole numbers. This means they can be 0, 1, 2, 3, etc.

3. **Non-Negative Integers**: The term "non-negative integers" includes all positive integers (1, 2, 3, ...) and the number 0. Hence, the exponents can be 0 or any positive integer.

4. **Examples of Polynomials**:
- \(3x^0\) is a polynomial because the exponent is 0.
- \(3x^2\) is a polynomial because the exponent is 2 (a positive integer).
- \(x^{-1}\) is NOT a polynomial because the exponent is -1 (a negative integer).

5. **Conclusion**: Therefore, in a polynomial, the exponents of the variables are always non-negative integers.

### Final Answer:
In a polynomial, the exponents of the variables are always non-negative integers.