Question

Jaime was asked to construct a polynomial function with the following
zeros and multiplicities. Which function should he write?
Zero Multiplicity
x=4 1
x= -2 3
f\left(x\right)=\left(x-4\right)\left(x+2\right)^3
f\left(x\right)=\left(x+4\right)\left(x-2\right)
f\left(x\right)=\left(x-4\right)\left(x+2\right)
f\left(x\right)=\left(x+4\right)\left(x-2\right)^3

Answer 1

To construct a polynomial function given the zeros and their multiplicities, we can use the following information:

1. A zero at \( x = 4 \) with a multiplicity of 1 means that the factor will be \( (x - 4) \).
2. A zero at \( x = -2 \) with a multiplicity of 3 means that the factor will be \( (x + 2)^3 \).

Combining these factors, we have:

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

Now let's evaluate the given choices:

1. \( f(x) = (x - 4)(x + 2)^3 \) - This one is correct.
2. \( f(x) = (x + 4)(x - 2) \) - This is incorrect; it does not match the zeros or their multiplicities.
3. \( f(x) = (x - 4)(x + 2) \) - This is incorrect; it does not account for the multiplicity of \( x = -2 \).
4. \( f(x) = (x + 4)(x - 2)^3 \) - This is incorrect; it does not match the specified zeros.

The correct polynomial function is:

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