Tatiana was asked to construct a polynomial function with the following zeros and multiplicities. Which function should she write? graph: zero| x = 1, x = 81 , Multiplicity| 3 , 5

Question

answer choices:
A
f(x)=(x+1)3(x+81)5f\left(x\right)=\left(x+1\right)^3\left(x+81\right)^5f(x)=(x+1)
3
(x+81)
5


B
f(x)=(x−1)5(x−81)3f\left(x\right)=\left(x-1\right)^5\left(x-81\right)^3f(x)=(x−1)
5
(x−81)
3


C
f(x)=(x−1)3(x−81)5f\left(x\right)=\left(x-1\right)^3\left(x-81\right)^5f(x)=(x−1)
3
(x−81)
5


D
f(x)=(x+1)5(x+81)3f\left(x\right)=\left(x+1\right)^5\left(x+81\right)^3f(x)=(x+1)
5
(x+81)
3

Answer 1

To construct the polynomial function given the zeros and their multiplicities, we follow these steps:

1. Identifying the zeros: The problem states the zeros \(x = 1\) and \(x = 81\).
2. Identifying the multiplicities: The multiplicity for \(x = 1\) is 3, and the multiplicity for \(x = 81\) is 5.

From this information, we can express the polynomial function:

• For the zero \(x = 1\) with multiplicity 3, we write this as \((x - 1)^3\).
• For the zero \(x = 81\) with multiplicity 5, we write this as \((x - 81)^5\).

Thus, the polynomial function \(f(x)\) should be:

\[ f(x) = (x - 1)^3(x - 81)^5 \]

Looking at the answer choices provided, we can find that:

• Choice C correctly reflects this polynomial:

\[ f(x) = (x - 1)^3(x - 81)^5 \]

Therefore, the answer is C.