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 $\frac{1}{6}$.

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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!