a polynomial function has a root of -4 with multiplicity 4, a root of -1 with multiplicity 3, and a root of…

Answer:

We need to analyze the behavior of the polynomial at its roots to determine the graph. First, find the degree of the polynomial. The degree is the sum of the multiplicities of its roots. The multiplicity of the root - 4 is 4, the multiplicity of the root - 1 is 3, and the multiplicity of the root 5 is 6. The degree (n=4 + 3+6=13) (an odd number as given).

1. Behavior at (x = - 4):

   • Since the multiplicity of the root (x=-4) is 4 (an even number), the graph of the polynomial touches the (x -)axis at (x = - 4).

2. Behavior at (x=-1):

   • Since the multiplicity of the root (x = - 1) is 3 (an odd number), the graph of the polynomial crosses the (x -)axis at (x=-1).

3. Behavior at (x = 5):

   • Since the multiplicity of the root (x = 5) is 6 (an even number), the graph of the polynomial touches the (x -)axis at (x = 5).

4. End - behavior:

   • Because the leading coefficient is positive and the degree of the polynomial is odd ((n = 13)), as (x\to-\infty), (y\to-\infty) and as (x\to+\infty), (y\to+\infty).

We would look for a graph that has the above - described behavior at (x=-4), (x = - 1), and (x = 5) and has the correct end - behavior.

However, since no options are provided, we can't give a specific choice. But the general characteristics of the correct graph are as described above.

Explanation:

Step1: Calculate the degree

The degree of the polynomial is the sum of multiplicities. (n=4 + 3+6=13).

Step2: Analyze behavior at (x=-4)

Even multiplicity means graph touches (x -)axis.

Step3: Analyze behavior at (x=-1)

Odd multiplicity means graph crosses (x -)axis.

Step4: Analyze behavior at (x = 5)

Even multiplicity means graph touches (x -)axis.

Step5: Determine end - behavior

Positive leading coefficient and odd degree: (x\to-\infty,y\to-\infty); (x\to+\infty,y\to+\infty).