The given function is:

$g(x) = 6x^7 - \pi x^4 + \frac{1}{5}x^3$

Is this a polynomial function?

A polynomial function is defined as a function of the form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

where $a_n, a_{n-1}, \dots, a_1, a_0$ are constants, and $n$ is a non-negative integer (degree of the polynomial).

In the given function:

• $6x^7$ is a term with degree 7.
• $\pi x^4$ is a term with degree 4 (note that $\pi$ is a constant).
• $\frac{1}{5}x^3$ is a term with degree 3.

Since all terms have non-negative integer powers of $x$, the function is a polynomial.

Degree of the polynomial:

The degree of a polynomial is the highest exponent of $x$. In this case, the highest exponent is 7, so the degree of the polynomial is 7.

Would you like more details or explanations?

Related Questions:

1. How do you determine whether a function is a polynomial?
2. What is the degree of the polynomial $3x^5 + 4x^2 - 7x + 9$?
3. What happens if a term in a function has a negative exponent of $x$?
4. What is the significance of the leading coefficient in a polynomial function?
5. Can constants like $\pi$ and $\frac{1}{5}$ be part of polynomial coefficients?

Tip:

For any polynomial, the degree is always determined by the term with the highest power of $x$, regardless of the coefficients.