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

Question

zero x=-5 , x=-10
multiplicity 2,5



A
f(x)=(x−5)2(x−10)5f\left(x\right)=\left(x-5\right)^2\left(x-10\right)^5f(x)=(x−5)
2
(x−10)
5


B
f(x)=(x+5)2(x+10)5f\left(x\right)=\left(x+5\right)^2\left(x+10\right)^5f(x)=(x+5)
2
(x+10)
5


C
f(x)=(x+5)5(x+10)2f\left(x\right)=\left(x+5\right)^5\left(x+10\right)^2f(x)=(x+5)
5
(x+10)
2


D
f(x)=(x−5)5(x−10)2f\left(x\right)=\left(x-5\right)^5\left(x-10\right)^2f(x)=(x−5)
5
(x−10)
2

Answer 1

To construct the polynomial function given the zeros and their corresponding multiplicities, we need to look at the zeros and their signs:

1. The zeros mentioned are \( x = -5 \) with a multiplicity of 2 and \( x = -10 \) with a multiplicity of 5.

This means that the factors of the polynomial can be expressed as:

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

Thus, the polynomial function should be written as: \[ f(x) = (x + 5)^2 (x + 10)^5 \]

Now, let's determine the correct answer option based on this factorization:

A: \( f(x) = (x - 5)^2 (x - 10)^5 \) - This is incorrect. B: \( f(x) = (x + 5)^2 (x + 10)^5 \) - This is correct. C: \( f(x) = (x + 5)^5 (x + 10)^2 \) - This is incorrect. D: \( f(x) = (x - 5)^5 (x - 10)^2 \) - This is incorrect.

Therefore, Trayvon should write: B: \( f(x) = (x + 5)^2 (x + 10)^5 \).