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

Question

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 determine which polynomial function Trayvon should write, we need the correct zeros and their corresponding multiplicities.

The given options provide different combinations of factors, which represent the zeros and their respective multiplicities.

From the polynomial functions listed, we should identify the signs and the degrees of multiplicity associated with each zero:

• If the zero is positive, it should correspond to a factor of the form \((x - a)\).
• If the zero is negative, it should correspond to a factor of the form \((x + a)\).

For each of the options provided:

A. \(f(x) = (x-5)^2(x-10)^5\)

• Zeros: \(x = 5\) (multiplicity 2), \(x = 10\) (multiplicity 5)

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

• Zeros: \(x = -5\) (multiplicity 2), \(x = -10\) (multiplicity 5)

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

• Zeros: \(x = -5\) (multiplicity 5), \(x = -10\) (multiplicity 2)

D. \(f(x) = (x-5)^5(x-10)^2\)

• Zeros: \(x = 5\) (multiplicity 5), \(x = 10\) (multiplicity 2)

To choose the correct function, it really depends on the specified zeros and their respective multiplicities that were given to Trayvon. However, since that information isn't specified here, let's summarize what we've analyzed:

• If the zeros are 5 with multiplicity 2 and 10 with multiplicity 5, then Option A is correct.
• If the zeros are -5 with multiplicity 2 and -10 with multiplicity 5, then Option B is correct.
• If the zeros are -5 with multiplicity 5 and -10 with multiplicity 2, then Option C is correct.
• If the zeros are 5 with multiplicity 5 and 10 with multiplicity 2, then Option D is correct.

Please clarify the specific zeros and multiplicities Trayvon was asked to work with to select the right answer.