Question

Trayvon was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?
\begin{array}{|c|c|} \hline \text{Zero} & \text{Multiplicity} \ \hline x = -5 & 2 \ \hline x = -10 & 5 \ \hline \end{array}

A. f(x)=(x+5)
2
(x+10)
5
B. f(x)=(x−5)
5
(x−10)
2
C. f(x)=(x+5)
5
(x+10)
2
D. f(x)=(x+5)
2
(x−10)
5

Answer 1

To construct a polynomial function with specified zeros and their multiplicities, we use the property that if \( x = a \) is a zero of multiplicity \( n \), then the factor corresponding to this zero will take the form \( (x - a)^n \).

Given the zeros and their multiplicities from your table:

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

Therefore, the polynomial function can be written as follows:

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

Now, let's look at the provided options:

A. \( f(x) = (x + 5)^2 (x + 10)^5 \) ✅
B. \( f(x) = (x - 5)^5 (x - 10)^2 \)
C. \( f(x) = (x + 5)^5 (x + 10)^2 \)
D. \( f(x) = (x + 5)^2 (x - 10)^5 \)

The correct choice is:

**A. f(x)=(x+5)²(x+10)⁵**