Question

Jaime was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write? zero:x=4,x=-2 multiplicity:1,3

Answer 1

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

- The zero \( x = 4 \) has a multiplicity of 1, which means it contributes a factor of \( (x - 4)^1 = (x - 4) \).
- The zero \( x = -2 \) has a multiplicity of 3, which means it contributes a factor of \( (x + 2)^3 \).

Now, we can combine these factors to construct the polynomial function:

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

Next, we can expand \( (x + 2)^3 \):

\[
(x + 2)^3 = (x + 2)(x + 2)(x + 2)
\]

Calculating \( (x + 2)(x + 2) \):

\[
(x + 2)(x + 2) = x^2 + 4x + 4
\]

Now multiply that by \( (x + 2) \):

\[
(x^2 + 4x + 4)(x + 2) = x^3 + 2x^2 + 4x^2 + 8x + 4 = x^3 + 6x^2 + 8x + 4
\]

So, \( (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \).

Now we can set up the full polynomial:

\[
f(x) = (x - 4)(x^3 + 6x^2 + 12x + 8)
\]

Next, we can expand this as follows:

1. Distributing \( (x - 4) \) to \( (x^3 + 6x^2 + 12x + 8) \):

\[
= x(x^3 + 6x^2 + 12x + 8) - 4(x^3 + 6x^2 + 12x + 8)
\]
\[
= x^4 + 6x^3 + 12x^2 + 8x - 4x^3 - 24x^2 - 48x - 32
\]

2. Combine like terms:

\[
= x^4 + (6x^3 - 4x^3) + (12x^2 - 24x^2) + (8x - 48x) - 32
\]
\[
= x^4 + 2x^3 - 12x^2 - 40x - 32
\]

So, the polynomial function Jaime should write is:

\[
f(x) = x^4 + 2x^3 - 12x^2 - 40x - 32
\]