The function given in the image is:

$f(x) = \frac{x^2 + 5}{x^3}$

To determine whether this is a polynomial, let's simplify the expression.

Simplifying the function:

$f(x) = \frac{x^2 + 5}{x^3} = \frac{x^2}{x^3} + \frac{5}{x^3} = x^{-1} + 5x^{-3}$

A polynomial is an expression that consists of terms with non-negative integer exponents of the variable $x$ (like $x^2$, $x^3$, etc.).

Analysis:

In the simplified function $f(x) = x^{-1} + 5x^{-3}$, both terms involve negative exponents ($-1$ and $-3$), which means this function is not a polynomial.

Conclusion:

The correct choice is:

• "It is not a polynomial."

Would you like further explanation on polynomials or exponents?

Here are 5 related questions to expand this topic:

1. What are the characteristics of a polynomial function?
2. How do you identify the degree of a polynomial function?
3. Can a function with negative exponents be a polynomial under any circumstances?
4. How would this function differ if all exponents were non-negative?
5. What is the impact of fractional exponents on polynomial classification?

Tip: Always check if the exponents of the variables are non-negative integers to determine if a function is a polynomial!