To find the degree of the polynomial function $P(x) = -(x - 3)^4(x - 8)(x + 9)(x + 1)$, we need to determine the total sum of the exponents of all the factors in the polynomial.

Step-by-Step Breakdown:

1. The term $(x - 3)^4$ has an exponent of 4.
2. The term $(x - 8)$ has an exponent of 1 (since it’s a linear factor).
3. The term $(x + 9)$ has an exponent of 1.
4. The term $(x + 1)$ has an exponent of 1.

Now, sum all the exponents: $4 + 1 + 1 + 1 = 7$

Conclusion:

The degree of the polynomial $P(x)$ is 7.

Would you like a detailed explanation of how each part contributes to the degree or have any other questions?

Here are 5 questions related to this topic:

1. How does the degree of a polynomial affect its graph?
2. What is the leading term of this polynomial function?
3. How do the roots of the polynomial relate to its factors?
4. Can the sign of the polynomial (positive or negative) affect its behavior?
5. What is the behavior of a 7th-degree polynomial at infinity?

Tip: The degree of a polynomial determines how many times its graph can intersect the x-axis and influences the end behavior of the function.