1. Identify the degree, leading term and leading coefficient of each polynomial function.

A. f(x)= x(x+1)(3x+1)(x-2)
B. f(x)= -16+3x^4 - 9x^2 - x^6 + 4x^8

2. Describe the end behavior of a ninth-degree polynomial function with a negative leading coefficient.

A degree 4, leading coeff 3

B degree 8, leading coeff 4

To identify the degree, leading term, and leading coefficient of each polynomial function, we need to understand the definitions of these terms.

1. Degree: The degree of a polynomial function is the highest exponent of the variable in the polynomial. It determines the shape and behavior of the graph.
2. Leading Term: The leading term is the term with the highest degree in the polynomial function.
3. Leading Coefficient: The leading coefficient is the coefficient of the leading term of the polynomial.

Let's apply this information to the given polynomial functions:

1. A. f(x) = x(x + 1)(3x + 1)(x - 2)
Degree: To find the degree, we determine the highest exponent. In this case, the highest exponent is 2x^2, which is derived from the term (3x + 1)(x - 2). Therefore, the degree of f(x) is 2.
Leading Term: The leading term is the term with the highest degree. In this case, it is 2x^2.
Leading Coefficient: The leading coefficient is the coefficient of the leading term. In this case, it is 1.

So, for function A:
Degree = 2
Leading Term = 2x^2
Leading Coefficient = 1

1. B. f(x) = -16 + 3x^4 - 9x^2 - x^6 + 4x^8
Degree: The highest exponent in this polynomial is 8, which comes from the term 4x^8. Hence, the degree of f(x) is 8.
Leading Term: The leading term is the term with the highest degree. In this case, it is 4x^8.
Leading Coefficient: The leading coefficient is the coefficient of the leading term. Here, it is 4.

So, for function B:
Degree = 8
Leading Term = 4x^8
Leading Coefficient = 4

Now, let's move on to the second question:

2. To understand the end behavior of a polynomial function, we need to consider the behavior of the graph as x approaches positive infinity and negative infinity.

Given that the ninth-degree polynomial function has a negative leading coefficient, we know that:

- As x approaches positive infinity, the function will decrease without bound.
- As x approaches negative infinity, the function will increase without bound.

This is because the negative leading coefficient causes the graph to "open" downwards, making the value of the function decrease as x moves towards positive infinity and increase as x moves towards negative infinity.

Thus, the end behavior for a ninth-degree polynomial function with a negative leading coefficient is a decreasing function as x approaches positive infinity and an increasing function as x approaches negative infinity.