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Part 1
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 1 minus x Superscript 4 Over 6 EndFraction
Question content area bottom
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.

Let's break this down step by step.

The given function is:

\[
g(x) = \frac{1 - x^4}{6}
\]

### Step 1: Determine if \( g(x) \) is a polynomial function
A polynomial function is a function of the form:

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

where the powers of \(x\) are non-negative integers, and the coefficients \(a_n, a_{n-1}, \dots, a_1, a_0\) are real numbers. Also, the exponents of \(x\) must be non-negative integers.

In the given function, the expression \( \frac{1 - x^4}{6} \) can be rewritten as:

\[
g(x) = \frac{1}{6} - \frac{1}{6}x^4
\]

### Step 2: Is \( g(x) \) a polynomial?
Yes, \( g(x) \) is a polynomial because:
- It is expressed as a sum of terms involving \(x\) raised to non-negative integer powers (in this case, \(x^0\) and \(x^4\)).
- The coefficients are constants (in this case, \( \frac{1}{6} \)).

### Step 3: Write the polynomial in standard form
Standard form arranges the polynomial in descending order of the powers of \(x\):

\[
g(x) = -\frac{1}{6}x^4 + \frac{1}{6}
\]

### Step 4: Identify the degree, leading term, and constant term
- **Degree**: The degree of the polynomial is the highest power of \(x\), which is \(4\).
- **Leading term**: The leading term is the term with the highest degree, which is \( -\frac{1}{6}x^4 \).
- **Constant term**: The constant term is the term without any \(x\) (i.e., the term with \(x^0\)), which is \( \frac{1}{6} \).

### Final Answer
- The function is a polynomial.
- The degree of the polynomial is 4.
- The polynomial in standard form is \( g(x) = -\frac{1}{6}x^4 + \frac{1}{6} \).
- The leading term is \( -\frac{1}{6}x^4 \).
- The constant term is \( \frac{1}{6} \).

---

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. What conditions must a function meet to be classified as a polynomial?
2. How do you identify the degree of a polynomial function?
3. Can a polynomial have negative exponents or fractions as exponents?
4. What is the role of the leading coefficient in a polynomial?
5. How does the degree of a polynomial affect its graph?

**Tip:** Always check the exponents and coefficients when determining if a function is 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) = (1 - x^4) / 6

Learn how to determine whether the function g(x) = (1 - x^4) / 6 is a polynomial function. This problem explains key polynomial concepts, such as identifying the degree, leading term, and constant term. Suitable for students in Grades 9-11.

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