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 minus 5 right parenthesis superscript 5 baseline left parenthesis x minus 10 right parenthesis squared 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)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)5(x+10)2

Answer 1

To construct a polynomial function based on the given zeros and their multiplicities, we start by understanding that zeros correspond to values of \( x \) that make the function equal to zero.

The given zeros and their multiplicities are:
- \( x = -5 \) with multiplicity 2
- \( x = -10 \) with multiplicity 5

When we express the polynomial in factored form, each zero \( x = r \) with multiplicity \( m \) contributes a factor \( (x - r)^m \).

For the zero \( x = -5 \) with multiplicity 2, the factor will be:
\[
(x + 5)^2
\]

For the zero \( x = -10 \) with multiplicity 5, the factor will be:
\[
(x + 10)^5
\]

Thus, the polynomial function is:
\[
f(x) = (x + 5)^2 (x + 10)^5
\]

This matches option:
- \( f(x) = (x + 5)^2 (x + 10)^5 \)

The correct response is:
\( f(x) = (x + 5)^2 (x + 10)^5 \)