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 seven halves Baseline minus x Superscript 4 Baseline plus 9
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Part 1
Determine whether​ g(x) is a polynomial or not. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
It is a polynomial of degree

enter your response here.
​(Type an integer or a​ fraction.)
B.
It is not a polynomial because the variable x is raised to the

enter your response here ​power, which is not a nonnegative integer.
​(Type an integer or a​ fraction.)
C.
It is not a polynomial because the function is the ratio of two distinct​ polynomials, and the polynomial in the denominator is of positive degree.

Let's analyze the given function \( g(x) = x^{7/2} - x^4 + 9 \).

### Step 1: Determine if the function is a polynomial
A **polynomial function** has the general form:

\[
P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
\]

where:
- The exponents of \(x\) must be **nonnegative integers** (i.e., 0, 1, 2, 3,...).
- The coefficients \(a_n, a_{n-1}, \dots, a_0\) are real numbers.

#### Analyzing each term:
- \( x^{7/2} \) has an exponent of \( 7/2 \), which is **not a nonnegative integer**.
- \( -x^4 \) has an exponent of 4, which is a nonnegative integer.
- The constant term \(9\) can be viewed as \(9x^0\), and the exponent 0 is also a nonnegative integer.

Because the first term \(x^{7/2}\) has an exponent that is not a nonnegative integer, this **disqualifies the function from being a polynomial**.

### Step 2: Conclusion
Since the variable \(x\) is raised to the power of \(7/2\) (which is not a nonnegative integer), this function is **not a polynomial**.

Thus, the correct answer is:

**B. It is not a polynomial because the variable \(x\) is raised to the \(7/2\) power, which is not a nonnegative integer.**

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Would you like further clarification on this topic? Here are some related questions to help you deepen your understanding:

1. What are the basic properties of a polynomial function?
2. Can a fractional exponent ever result in a polynomial?
3. How do you find the degree of a polynomial?
4. What distinguishes a monomial from a polynomial?
5. How do you identify the leading term of a polynomial?

**Tip**: A polynomial can have only whole-number exponents (including zero). Any fractional or negative exponent will disqualify it from being a polynomial.

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) equals x^(7/2) minus x^4 plus 9.

Explore whether the function g(x) = x^(7/2) - x^4 + 9 is a polynomial. This problem covers key concepts such as polynomials, exponents, and the standard form of a polynomial. Suitable for students in Grades 9-12.

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