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)equalsStartFraction 4 minus x cubed Over 3 EndFraction
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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:

\[
g(x) = \frac{4 - x^3}{3}
\]

This function can be rewritten as:

\[
g(x) = \frac{1}{3}(4 - x^3) = \frac{1}{3} \cdot 4 - \frac{1}{3} \cdot x^3 = \frac{4}{3} - \frac{1}{3}x^3
\]

### Step 1: Check if it's a polynomial function

A **polynomial function** is an expression made up of terms, each consisting of a constant coefficient and a variable raised to a non-negative integer exponent. There must be no division by variables.

In this case, after simplifying the expression for \( g(x) \), the terms are:

- \( \frac{4}{3} \), a constant term.
- \( -\frac{1}{3}x^3 \), a term with \( x \) raised to a non-negative integer power (3).

There is no division by \( x \), and both terms involve constants or powers of \( x \) that are non-negative integers. Therefore, this is a polynomial function.

### Step 2: Identify the degree

The **degree** of a polynomial function is the highest power of the variable \( x \) in the expression. In this case, the highest power of \( x \) is \( 3 \).

Thus, the degree of this polynomial is 3.

### Step 3: Write the polynomial in standard form

The standard form of a polynomial is written with the terms in decreasing order of the exponents of \( x \):

\[
g(x) = -\frac{1}{3}x^3 + \frac{4}{3}
\]

### Step 4: Identify the leading term and the constant term

- The **leading term** is the term with the highest power of \( x \), which is \( -\frac{1}{3}x^3 \).
- The **constant term** is the term without \( x \), which is \( \frac{4}{3} \).

### Conclusion:

The function \( g(x) \) is a polynomial of degree 3. The leading term is \( -\frac{1}{3}x^3 \), and the constant term is \( \frac{4}{3} \).

---

Would you like further details on this or any related topic?

Here are 5 related questions:
1. What makes a function not a polynomial?
2. How do you determine the degree of more complex polynomial functions?
3. Can polynomial functions have negative exponents or fractional exponents?
4. How does the degree of a polynomial affect its graph's behavior?
5. What are some examples of non-polynomial functions?

**Tip**: Always ensure that variables in a polynomial are raised to non-negative integer exponents and not in the denominator.

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) = (4 - x^3) / 3

Analyze the function g(x) = (4 - x^3) / 3 to determine whether it's a polynomial. Learn how to rewrite the function in standard form, identify the degree, and find the leading and constant terms. Suitable for high school students learning about polynomials.

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