Certainly! Let's find the polynomial [tex]\( f(x) \)[/tex] that has the given roots: [tex]\(-4\)[/tex], [tex]\(4i\)[/tex], and [tex]\(-4i\)[/tex].
1. Start with the factors corresponding to the given roots:
- If a polynomial has a root [tex]\( r \)[/tex], then [tex]\( f(x) \)[/tex] will have a factor [tex]\( (x - r) \)[/tex].
Therefore, the polynomial [tex]\( f(x) \)[/tex] will have the factors [tex]\( (x + 4) \)[/tex], [tex]\( (x - 4i) \)[/tex], and [tex]\( (x + 4i) \)[/tex].
2. Multiply these factors to form the polynomial:
- First, multiply the factors with the imaginary roots:
[tex]\[
(x - 4i)(x + 4i)
\][/tex]
This is a difference of squares, which simplifies to:
[tex]\[
(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 - (-16) = x^2 + 16
\][/tex]
3. Now, incorporate the real root factor:
- Multiply the result by the factor [tex]\( (x + 4) \)[/tex]:
[tex]\[
(x + 4)(x^2 + 16)
\][/tex]
4. Expand the expression:
- Distribute [tex]\( (x + 4) \)[/tex] across [tex]\( (x^2 + 16) \)[/tex]:
[tex]\[
(x + 4)(x^2 + 16) = x(x^2 + 16) + 4(x^2 + 16)
\][/tex]
This becomes:
[tex]\[
x^3 + 16x + 4x^2 + 64
\][/tex]
5. Rearrange terms in standard polynomial form:
- The resulting polynomial is:
[tex]\[
f(x) = x^3 + 4x^2 + 16x + 64
\][/tex]
Thus, the polynomial [tex]\( f(x) \)[/tex] that has roots [tex]\(-4\)[/tex], [tex]\(4i\)[/tex], and [tex]\(-4i\)[/tex] is:
[tex]\[
f(x) = x^3 + 4x^2 + 16x + 64
\][/tex]
