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

Question

table zero x=1
x=81
multiply
3
5



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 determine the correct polynomial function based on the given zeros and their multiplicities, we need to carefully analyze the information provided:

• Zero \( x = 1 \) with multiplicity \( 3 \)
• Zero \( x = 81 \) with multiplicity \( 5 \)

This means:

• For the zero \( x = 1 \), the factor will be \( (x-1) \) raised to the power of the multiplicity, which is \( 3 \). Hence, the factor is \( (x-1)^3 \).

• For the zero \( x = 81 \), the factor will be \( (x-81) \) raised to the power of the multiplicity, which is \( 5 \). Hence, the factor is \( (x-81)^5 \).

Now, we can combine these factors to construct the polynomial function as follows:

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

Looking at the options provided:

A. \( f(x) = (x + 1)^3 (x + 81)^5 \)
B. \( f(x) = (x - 1)^5 (x - 81)^3 \)
C. \( f(x) = (x - 1)^3 (x - 81)^5 \)
D. \( f(x) = (x + 1)^5 (x + 81)^3 \)

The correct option following our construction is:

C: \( f(x) = (x - 1)^3 (x - 81)^5 \)