The function given is:

$g(x) = \frac{6 - x^6}{5}$

Step 1: Check if the function is a polynomial

A function is a polynomial if:

1. All exponents of the variable $x$ are non-negative integers.
2. The function does not involve division by a variable (denominators must not contain variables).

In this case, $g(x)$ is of the form $\frac{6 - x^6}{5}$, which simplifies to: $g(x) = \frac{6}{5} - \frac{1}{5}x^6$

This is a valid polynomial because:

• The exponents of $x$ (specifically, 6 in $\frac{1}{5}x^6$) are non-negative integers.
• There is no division by a variable (the division by 5 is a constant).

Step 2: Degree of the polynomial

The degree of a polynomial is the highest power of $x$. In this case, the highest power of $x$ is $6$, so the degree of the polynomial is 6.

Step 3: Identify the leading term and constant term

• The leading term is the term with the highest power of $x$, which is $-\frac{1}{5}x^6$.
• The constant term is the term without $x$, which is $\frac{6}{5}$.

Answer Choices:

• Option A is correct: "It is a polynomial of degree 6."

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you determine if a function with a fraction is a polynomial?
2. What is the general form of a polynomial?
3. Can a polynomial have negative exponents?
4. How does dividing by a constant affect whether a function is a polynomial?
5. What happens to the degree of a polynomial if all terms have the same degree?

Tip: The leading term of a polynomial always determines its behavior for large values of $x$.