https://file.aiquickdraw.com/mathbot/file/1729468234446v0ewdibl.jpg

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?

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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!

Determine if the following function is a polynomial function. If it is, identify the degree. f(x) = (x^2 + 5) / x^3

Learn how to determine if the function f(x) = (x^2 + 5) / x^3 is a polynomial. This problem explores key algebraic concepts such as simplifying polynomial fractions and understanding the definition of polynomial functions. Suitable for students in Grades 9-12.

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