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Part 1

Determine whether the function is a polynomial function. If it​ is, identify the degree.

​g(x)equals=6 x Superscript 7 Baseline minus pi x Superscript 4 Baseline plus one fifth x cubed6x7−πx4+15x3




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.

Determine whether the function is a polynomial function. If it is, identify the degree. g(x) = 6x^7 − πx^4 + 1/5x^3

Analyze the function g(x) = 6x^7 − πx^4 + 1/5x^3 to determine if it qualifies as a polynomial. Understand key concepts like polynomial structure, the role of constants like π and fractions, and how to identify the degree of a polynomial function. Suitable for students in Grades 10-12.

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