Determine whether the following function is a polynomial function. If the function is a polynomial​ function, state its degree. If it is​ not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term.
​g(x)equalsx Superscript five halves Baseline minus x squared plus 8

Let's analyze the given function:

\[ g(x) = x^{\frac{5}{2}} - x^2 + 8 \]

### Step 1: Is it a polynomial function?

A polynomial function is one where all exponents of the variable \( x \) are **non-negative integers**. Let's examine each term:

- The term \( x^{\frac{5}{2}} \) has an exponent that is not an integer (it's a fraction), which violates the definition of a polynomial function.
- The term \( -x^2 \) is fine because 2 is a non-negative integer.
- The constant term \( 8 \) is also acceptable.

However, since \( x^{\frac{5}{2}} \) involves a fractional exponent, **this function is not a polynomial function**.

### Step 2: Why it is not a polynomial?

The function is not a polynomial because the first term, \( x^{\frac{5}{2}} \), has a fractional exponent. Polynomial functions must have exponents that are whole numbers (integers).

### Conclusion:

The function \( g(x) = x^{\frac{5}{2}} - x^2 + 8 \) is **not a polynomial** due to the fractional exponent in \( x^{\frac{5}{2}} \). Therefore, no degree, leading term, or constant term can be identified under the context of polynomial functions.

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Would you like further clarification on any part of this, or want to explore related topics?

Here are some related questions for further study:
1. What are the properties of polynomial functions?
2. How do you determine the degree of a polynomial function?
3. What makes a term in a function violate the polynomial criteria?
4. Can a function with rational exponents be part of a different category of functions?
5. How do constant terms affect the degree of a polynomial function?

**Tip:** Always check for non-integer or negative exponents when identifying polynomial functions.

Determine whether the following function is a polynomial function. If the function is a polynomial​ function, state its degree. If it is​ not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term. ​g(x) = x^(5/2) - x^2 + 8

Learn how to determine if the function g(x) = x^(5/2) - x^2 + 8 is a polynomial function. This problem explores key algebraic concepts, such as the definition of polynomial functions and the role of exponents in determining whether a function qualifies as a polynomial. Suitable for Grades 10-12.

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