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 $a = -4$.

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 $x$.