Certainly! Let's start by understanding that the roots of a polynomial with real coefficients must come in complex conjugate pairs. This means if [tex]\(2 + i\)[/tex] is a root, then [tex]\(2 - i\)[/tex] must also be a root.
Given the roots [tex]\(2 + i\)[/tex], [tex]\(5\)[/tex], and we can include its conjugate [tex]\(2 - i\)[/tex], we need to form the polynomial using these roots.
1. The polynomial can be expressed in terms of its factors corresponding to the roots:
[tex]\[
(x - (2 + i))(x - (2 - i))(x - 5)
\][/tex]
2. We can simplify this step-by-step. First, simplify the product of the complex conjugate pair:
[tex]\[
(x - (2 + i))(x - (2 - i))
\][/tex]
3. Use the difference of squares formula to simplify this:
[tex]\[
(x - 2 - i)(x - 2 + i) = (x - 2)^2 - i^2
\][/tex]
4. Since [tex]\(i^2 = -1\)[/tex], the expression becomes:
[tex]\[
(x - 2)^2 + 1
\][/tex]
5. Thus, the polynomial is:
[tex]\[
((x - 2)^2 + 1)(x - 5)
\][/tex]
Next, identify the missing root:
Given the roots are [tex]\(2 + i\)[/tex], [tex]\(5\)[/tex], and the conjugate root [tex]\(2 - i\)[/tex].
Therefore, the additional root must be:
[tex]\[
2 - i
\][/tex]
Among the options provided:
- [tex]\(-3\)[/tex]
- [tex]\(-5\)[/tex]
- [tex]\(2 - i\)[/tex]
- [tex]\(2i\)[/tex]
The correct additional root is:
[tex]\[
2 - i
\][/tex]
Thus, the polynomial must also have the root: [tex]\(\boxed{2 - i}\)[/tex].
