To determine which of the given expressions is a polynomial, we need to analyze each expression to see if they meet the criteria for being a polynomial. A polynomial is an expression that consists of variables and coefficients, where the variables have non-negative integer exponents.
Let's analyze each expression step-by-step:
1. Expression: [tex]\( 9x^7 y^{-3} z \)[/tex]
- Here, [tex]\( x \)[/tex] has an exponent of 7, which is a non-negative integer.
- However, [tex]\( y \)[/tex] has an exponent of -3, which is a negative integer.
- Recall that for an expression to be a polynomial, all exponents must be non-negative integers. Because [tex]\( y \)[/tex] has a negative exponent, this expression is NOT a polynomial.
2. Expression: [tex]\( 4x^3 - 2x^2 + 5x - 6 + \frac{1}{x} \)[/tex]
- Initially, it seems like a polynomial with terms [tex]\( 4x^3 \)[/tex], [tex]\( -2x^2 \)[/tex], [tex]\( 5x \)[/tex], and [tex]\( -6 \)[/tex] (constant term), all of which have non-negative integer exponents.
- However, the term [tex]\(\frac{1}{x}\)[/tex] can be rewritten as [tex]\( x^{-1} \)[/tex], which has a negative exponent.
- Because of the term [tex]\( x^{-1} \)[/tex], this expression is NOT a polynomial.
3. Expression: [tex]\( -13 \)[/tex]
- This is a constant term.
- A constant term is considered a polynomial of degree 0, since it can be written as [tex]\( -13x^0 \)[/tex], where 0 is a non-negative integer.
- Thus, this expression IS a polynomial.
4. Expression: [tex]\( 13x^{-2} \)[/tex]
- Here, [tex]\( x \)[/tex] has an exponent of -2, which is a negative integer.
- For an expression to be a polynomial, all exponents must be non-negative integers.
- Because [tex]\( x \)[/tex] has a negative exponent of -2, this expression is NOT a polynomial.
Based on this analysis, the expression that is a polynomial is:
[tex]\[ -13 \][/tex]
