Tatiana was asked to construct a polynomial function with the following zeros and multiplicities. Which function should she write?

Question

(1 point)
Responses

f(x) = (x+1)3(x+81)5
f(x) = (x+1)3(x+81)5

f(x) = (x−1)5(x−81)3
f(x) = (x−1)5(x−81)3

f(x) = (x+1)5(x+81)3
f(x) = (x+1)5(x+81)3

f(x) = (x−1)3(x−81)5

Answer 1

To determine which polynomial function Tatiana should write, we need to identify the zeros and their corresponding multiplicities based on her given instructions.

We have four options, and we need to evaluate zeros and multiplicities from those.

1. Option 1: \( f(x) = (x+1)^3 (x+81)^5 \)

   • Zeros: -1 (multiplicity 3), -81 (multiplicity 5)

2. Option 2: \( f(x) = (x-1)^5 (x-81)^3 \)

   • Zeros: 1 (multiplicity 5), 81 (multiplicity 3)

3. Option 3: \( f(x) = (x+1)^5 (x+81)^3 \)

   • Zeros: -1 (multiplicity 5), -81 (multiplicity 3)

4. Option 4: \( f(x) = (x-1)^3 (x-81)^5 \)

   • Zeros: 1 (multiplicity 3), 81 (multiplicity 5)

Now, without any specifics on which zeros and multiplicities were requested from Tatiana, we cannot determine exactly which function she should construct. However, based on the representations:

• If she needed zeros at -1 and -81 with multiplicities of 3 and 5 respectively, she should choose Option 1.
• If she needed zeros at 1 and 81 with those multiplicities swapped, she should choose Option 2.
• If she needed zeros at -1 and -81 with multiplicities swapped, she would choose Option 3.
• If she needed zeros at 1 and 81 but in the given format, she should go for Option 4.

If no specific details are provided regarding the zeros and their multiplicities, she should confirm what exactly is required. Based on general polynomial construction principles and common choices, Option 1 is a reasonable choice with large multiplicities.