Question

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

Zero Multiplicity
x=−5 2
x=−10 5
(1 point)
Responses

f(x)=(x+5)5(x+10)2
f left parenthesis x right parenthesis equals left parenthesis x plus 5 right parenthesis superscript 5 baseline left parenthesis x plus 10 right parenthesis squared

f(x)=(x+5)2(x+10)5
f left parenthesis x right parenthesis equals left parenthesis x plus 5 right parenthesis squared left parenthesis x plus 10 right parenthesis superscript 5 baseline

f(x)=(x−5)2(x−10)5
f left parenthesis x right parenthesis equals left parenthesis x minus 5 right parenthesis squared left parenthesis x minus 10 right parenthesis superscript 5 baseline

f(x)=(x−5)5(x−10)2

Answer 1

To construct a polynomial function based on the provided zeros and their multiplicities, we start by translating the zeros x = -5 and x = -10 into factors:

1. For the zero at x = -5 with multiplicity 2, the corresponding factor is \((x + 5)^2\).
2. For the zero at x = -10 with multiplicity 5, the corresponding factor is \((x + 10)^5\).

Now, we need to combine these factors into a single polynomial function:

\[
f(x) = (x + 5)^2 (x + 10)^5
\]

Looking at the provided options, the correct function Trayvon should write is:

**f(x) = (x + 5)²(x + 10)⁵**.

Thus, the correct answer is:

**f(x) = (x + 5)²(x + 10)⁵**.