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 of x is equal to open paren x plus 5 close paren to the 5th power times open paren x plus 10 close paren squared

f(x)=(x−5)2(x−10)5
f of x is equal to open paren x minus 5 close paren squared times open paren x minus 10 close paren to the 5th power

f(x)=(x−5)5(x−10)2
f of x is equal to open paren x minus 5 close paren to the 5th power times open paren x minus 10 close paren squared

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

Answer 1

To construct a polynomial function based on the provided zeros and their multiplicities, we need to use the fact that:

- A zero of a polynomial at \( x = c \) means that \( (x - c) \) is a factor of the polynomial.
- The multiplicity indicates how many times that factor is repeated.

In this case, Trayvon has the following zeros and multiplicities:

- Zero \( x = -5 \) with multiplicity 2 implies the factor \( (x + 5)^2 \).
- Zero \( x = -10 \) with multiplicity 5 implies the factor \( (x + 10)^5 \).

Thus, the polynomial function that Trayvon should write is:

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

So the correct response from the options given is:

**f(x) = (x + 5)²(x + 10)⁵**