Question

Jaime was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?
\[<br />
\begin{array}{|c|c|}<br />
\hline<br />
\text{Zero} & \text{Multiplicity} \n<br />
\hline<br />
x = 4 & 1 \n<br />
\hline<br />
x = -2 & 3 \n<br />
\hline<br />
\end{array}<br />
\]

A. f(x) = (x-4)(x+2)^3
B. f(x) = (x+4)(x-2)
C. f(x) = (x-4)(x+2)
D. f(x) = (x+4)(x-2)^3

Answer 1

To construct a polynomial function given the zeros and their multiplicities, we use the fact that each zero \( r \) with multiplicity \( m \) contributes a factor of \( (x - r)^m \) to the polynomial.

From the table given:

- The zero \( x = 4 \) has a multiplicity of 1. Therefore, the factor contributed by this zero is \( (x - 4)^1 = (x - 4) \).
- The zero \( x = -2 \) has a multiplicity of 3. Therefore, the factor contributed by this zero is \( (x + 2)^3 \).

Putting these factors together, the polynomial function can be expressed as:

\[
f(x) = (x - 4)(x + 2)^3
\]

Now, let's match this with the provided options:

A. \( f(x) = (x-4)(x+2)^3 \) ✅

B. \( f(x) = (x+4)(x-2) \)

C. \( f(x) = (x-4)(x+2) \)

D. \( f(x) = (x+4)(x-2)^3 \)

The correct answer is:

**A. \( f(x) = (x-4)(x+2)^3 \)**.