To determine which expression is a polynomial, we need to review the properties of polynomials. A polynomial is an algebraic expression that can have constants, variables, and exponents, but the exponents must be non-negative integers, and the variables should not appear in the denominator.
Let's evaluate each given expression to see if they meet these criteria:
1. [tex]\( 9x^7y^{-3}z \)[/tex]
- This expression contains the term [tex]\( y^{-3} \)[/tex], which has a negative exponent.
- Polynomials do not allow negative exponents, so this expression is not a polynomial.
2. [tex]\( 4x^3 - 2x^2 + 5x - 6 + \frac{1}{x} \)[/tex]
- The terms [tex]\( 4x^3 \)[/tex], [tex]\( -2x^2 \)[/tex], [tex]\( 5x \)[/tex], and [tex]\( -6 \)[/tex] are all valid terms for a polynomial.
- However, the term [tex]\( \frac{1}{x} \)[/tex] contains a variable in the denominator.
- Polynomials do not have variables in the denominator, so this expression is not a polynomial.
3. [tex]\( -13 \)[/tex]
- This expression is a constant, which can be considered a polynomial of degree 0.
- It fits the definition of a polynomial, as it has no variables or exponents that violate the polynomial criteria.
- Therefore, this expression is a polynomial.
4. [tex]\( 13x^{-2} \)[/tex]
- This expression contains the term [tex]\( x^{-2} \)[/tex], which has a negative exponent.
- Polynomials do not allow negative exponents, so this expression is not a polynomial.
Based on the evaluation, the expressions that are polynomials are:
- [tex]\(-13\)[/tex]
Hence, the expression (c) [tex]\( -13 \)[/tex] is the polynomial.
