To create a polynomial function [tex]\( f(x) \)[/tex] of degree 3 that has the zeros [tex]\(4\)[/tex], [tex]\(2i\)[/tex], and [tex]\(-2i\)[/tex], we can follow these steps:
1. List the zeros: The given zeros are [tex]\( 4 \)[/tex], [tex]\( 2i \)[/tex], and [tex]\( -2i \)[/tex].
2. Form the factors: For each zero [tex]\( a \)[/tex], there is a factor [tex]\( (x - a) \)[/tex] in the polynomial. Therefore, the factors are:
[tex]\[
(x - 4), \, (x - 2i), \, (x + 2i)
\][/tex]
3. Multiply the factors: The next step is to multiply these factors together to form the polynomial.
Let's start by multiplying the complex conjugate factors [tex]\( (x - 2i) \)[/tex] and [tex]\( (x + 2i) \)[/tex] first:
[tex]\[
(x - 2i)(x + 2i)
\][/tex]
This is a difference of squares, which simplifies as follows:
[tex]\[
x^2 - (2i)^2 = x^2 - 4(-1) = x^2 + 4
\][/tex]
Now, we need to multiply this result by the remaining factor [tex]\( (x - 4) \)[/tex]:
[tex]\[
(x - 4)(x^2 + 4)
\][/tex]
Expand this product by distributing [tex]\( x - 4 \)[/tex] over [tex]\( x^2 + 4 \)[/tex]:
[tex]\[
(x - 4)(x^2 + 4) = x(x^2 + 4) - 4(x^2 + 4)
\][/tex]
Distribute each term:
[tex]\[
x^3 + 4x - 4x^2 - 16
\][/tex]
Arrange the terms in descending order of power:
[tex]\[
x^3 - 4x^2 + 4x - 16
\][/tex]
Thus, the polynomial [tex]\( f(x) \)[/tex] that has the zeros [tex]\(4\)[/tex], [tex]\(2i\)[/tex], and [tex]\(-2i\)[/tex] is:
[tex]\[
f(x) = x^3 - 4x^2 + 4x - 16
\][/tex]
This polynomial meets the given conditions.
