To find the polynomial function of lowest degree with lead coefficient 1 and roots [tex]\(i, -2\)[/tex], and [tex]\(2\)[/tex], we need to follow these steps:
1. Identify the given roots: The roots of the polynomial are [tex]\(i, -2\)[/tex], and [tex]\(2\)[/tex].
2. Include the complex conjugate: Since [tex]\(i\)[/tex] is a root of the polynomial, its complex conjugate [tex]\(-i\)[/tex] must also be a root. Therefore, the complete set of roots is [tex]\(i, -i, -2,\)[/tex] and [tex]\(2\)[/tex].
3. Form the factors: Each root gives a factor of the polynomial:
- For [tex]\(i\)[/tex], the factor is [tex]\((x - i)\)[/tex].
- For [tex]\(-i\)[/tex], the factor is [tex]\((x + i)\)[/tex].
- For [tex]\(-2\)[/tex], the factor is [tex]\((x + 2)\)[/tex].
- For [tex]\(2\)[/tex], the factor is [tex]\((x - 2)\)[/tex].
4. Multiply the factors in pairs: Expand the polynomial step-by-step by multiplying these factors.
- First, multiply the factors involving [tex]\(i\)[/tex]:
[tex]\[
(x - i)(x + i) = x^2 - i^2
\][/tex]
Note that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
(x - i)(x + i) = x^2 - (-1) = x^2 + 1
\][/tex]
- Next, multiply the factors involving [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[
(x + 2)(x - 2) = x^2 - 4
\][/tex]
5. Combine the results: Now, multiply the two quadratic expressions obtained:
[tex]\[
(x^2 + 1)(x^2 - 4)
\][/tex]
6. Expand the product:
[tex]\[
(x^2 + 1)(x^2 - 4) = x^2(x^2 - 4) + 1(x^2 - 4)
\][/tex]
Simplify the expression:
[tex]\[
x^2(x^2 - 4) + 1(x^2 - 4) = x^4 - 4x^2 + x^2 - 4 = x^4 - 3x^2 - 4
\][/tex]
Thus, the polynomial of the lowest degree with lead coefficient 1 and roots [tex]\(i, -i, -2\)[/tex], and [tex]\(2\)[/tex] is:
[tex]\[
f(x) = x^4 - 3x^2 - 4
\][/tex]
So the correct answer is:
[tex]\[
\boxed{f(x) = x^4 - 3 x^2 - 4}
\][/tex]
