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

Question

Zero Multiplicity
x=1
3
x=81
5(1 point)
Responses

f(x)=(x+1)5(x+81)3
f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis superscript 5 baseline left parenthesis x plus 81 right parenthesis cubed

f(x)=(x+1)3(x+81)5
f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis cubed left parenthesis x plus 81 right parenthesis superscript 5 baseline

f(x)=(x−1)3(x−81)5
f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis cubed left parenthesis x minus 81 right parenthesis superscript 5 baseline

f(x)=(x−1)5(x−81)3

Answer 1

To construct a polynomial function based on the given zeros and their multiplicities, we should use the facts that:

• For a zero at \( x = r \) with multiplicity \( m \), the corresponding factor in the polynomial will be \( (x - r)^m \).

Given the information:

1. Zero at \( x = 1 \) with multiplicity 3: This contributes a factor of \( (x - 1)^3 \).
2. Zero at \( x = 81 \) with multiplicity 5: This contributes a factor of \( (x - 81)^5 \).

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

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

Now let's examine the provided responses:

1. Response 1: \( f(x) = (x + 1)^5 (x + 81)^3 \) - Incorrect.
2. Response 2: \( f(x) = (x + 1)^3 (x + 81)^5 \) - Incorrect.
3. Response 3: \( f(x) = (x - 1)^3 (x - 81)^5 \) - Correct.
4. Response 4: \( f(x) = (x - 1)^5 (x - 81)^3 \) - Incorrect.

The correct choice is:

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