To determine which expression is a polynomial, we need to evaluate each option based on the definition of a polynomial. A polynomial is an algebraic expression that involves only non-negative integer exponents of the variables.
Let's analyze each given expression:
1. \(9x^7 y^{-3} z\):
- This expression contains the term \(y^{-3}\), which has a negative exponent.
- Polynomials do not include negative exponents.
- Therefore, \(9x^7 y^{-3} z\) is not a polynomial.
2. \(4x^3 - 2x^2 + 5x - 6 + \frac{1}{x}\):
- This expression includes the term \(\frac{1}{x}\), which can be rewritten as \(x^{-1}\).
- The term \(x^{-1}\) has a negative exponent.
- Therefore, \(4x^3 - 2x^2 + 5x - 6 + \frac{1}{x}\) is not a polynomial.
3. \(-13\):
- This expression is just a constant.
- A constant can be considered a polynomial of degree 0 (e.g., \(-13\) is equivalent to \(-13x^0\)).
- Therefore, \(-13\) is a polynomial.
4. \(13x^{-2}\):
- This expression contains the term \(x^{-2}\), which has a negative exponent.
- Polynomials do not include negative exponents.
- Therefore, \(13x^{-2}\) is not a polynomial.
Based on the analysis above, the only expression that satisfies the definition of a polynomial is:
\(-13\)
Thus, the correct answer is the third option.
