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

Question

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

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
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)5(x+10)2

Answer 1

To construct a polynomial function with specified zeros and their corresponding multiplicities, we should use the following approach:

1. Identify the zeros and their multiplicities:

   • Zero at \( x = -5 \) with multiplicity 2
   • Zero at \( x = -10 \) with multiplicity 5

2. Write the factors corresponding to each zero:

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

3. Combine the factors to form the polynomial:

   • The polynomial function will be \( f(x) = (x + 5)^2 (x + 10)^5 \).

Therefore, the correct function Trayvon should write is:

f(x) = (x + 5)^2 (x + 10)^5.