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

Question

Zero Multiplicity
x=4 1
x=−2 3(1 point)
Responses

f(x)=(x−4)(x+2)3
f left parenthesis x right parenthesis equals left parenthesis x minus 4 right parenthesis left parenthesis x plus 2 right parenthesis cubed

f(x)=(x+4)(x−2)
f left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis left parenthesis x minus 2 right parenthesis

f(x)=(x+4)(x−2)3
f left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis left parenthesis x minus 2 right parenthesis cubed

f(x)=(x−4)(x+2)

Answer 1

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

• A zero at \(x = 4\) with multiplicity 1 implies a factor of \((x - 4)^1\).
• A zero at \(x = -2\) with multiplicity 3 implies a factor of \((x + 2)^3\).

Therefore, the polynomial function can be constructed as follows:

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

Now, let's look at the responses provided:

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

The first option matches our derived function.

So, Jaime should write:

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