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Let's analyze the given function \( f(x) = 5 - 6x - 4x^4 \) to determine if it is a polynomial, and if it is, we will identify its degree and leading coefficient.

### Step 1: Is the function a polynomial?
A polynomial function is of the form:
\[
f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
\]
where \( n \) is a non-negative integer, and the coefficients \( a_n, a_{n-1}, \dots, a_1, a_0 \) are real numbers.

- In this case, the function is \( f(x) = 5 - 6x - 4x^4 \).
- The terms are \( 5 \), \( -6x \), and \( -4x^4 \), all of which are valid polynomial terms (with non-negative integer powers of \( x \)).

Thus, the function is a polynomial.

### Step 2: Degree of the polynomial
The degree of a polynomial is the highest power of \( x \) in the function.

- In \( f(x) = 5 - 6x - 4x^4 \), the term with the highest power of \( x \) is \( -4x^4 \).
- Therefore, the degree of the polynomial is \( 4 \).

### Step 3: Leading coefficient
The leading coefficient is the coefficient of the term with the highest degree.

- The term with the highest degree is \( -4x^4 \), and its coefficient is \( -4 \).
- Therefore, the leading coefficient \( a \) is \( -4 \).

### Conclusion
- The function is a polynomial with **degree 4**.
- The **leading coefficient** is \( -4 \).

### Answer:
Option A: The function is a polynomial function with the degree 4 and the leading coefficient \( a = -4 \).

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Would you like more details or have any questions?

Here are 5 additional questions you can consider:

1. What makes a function *not* a polynomial?
2. Can a polynomial have negative or fractional exponents?
3. How does the degree of a polynomial affect its graph?
4. What happens if a polynomial has a leading coefficient of zero?
5. What is the importance of the leading coefficient in determining the end behavior of a polynomial?

**Tip:** The degree of a polynomial helps you understand its general shape, while the leading coefficient affects how it behaves at extreme values of \( x \).

Determine if the function is a polynomial function. If it is, state its degree and leading coefficient a.
f(x) = 5 - 6x - 4x^4

Learn how to determine if a function is a polynomial and identify its degree and leading coefficient. This problem involves analyzing the polynomial f(x) = 5 - 6x - 4x^4, covering key concepts like the degree of a polynomial and the leading coefficient. Suitable for students in Grades 9-12.

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