To find the polynomial function with a leading coefficient of 1 and roots [tex]\(2i\)[/tex] and [tex]\(3i\)[/tex] each with a multiplicity of 1, we follow these steps:
1. Identify the given roots: The roots of the polynomial are [tex]\(2i\)[/tex] and [tex]\(3i\)[/tex] each with a multiplicity of 1. Roots also come with their complex conjugates, thus the full set of roots is [tex]\(\{2i, -2i, 3i, -3i\}\)[/tex].
2. Form the factors of the polynomial: For each root [tex]\(r\)[/tex], the corresponding factor is [tex]\((x - r)\)[/tex]. Therefore, the factors are:
[tex]\[
(x - 2i), (x + 2i), (x - 3i), (x + 3i)
\][/tex]
3. Construct the polynomial by multiplying these factors:
[tex]\[
f(x) = (x - 2i)(x + 2i)(x - 3i)(x + 3i)
\][/tex]
4. Simplify pairs of complex conjugate factors:
[tex]\[
(x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4
\][/tex]
Similarly,
[tex]\[
(x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9
\][/tex]
5. Multiply the resulting quadratic polynomials:
[tex]\[
f(x) = (x^2 + 4)(x^2 + 9)
\][/tex]
6. Expand the polynomial using the distributive property:
[tex]\[
f(x) = x^2(x^2 + 9) + 4(x^2 + 9)
\][/tex]
[tex]\[
= x^4 + 9x^2 + 4x^2 + 36
\][/tex]
[tex]\[
= x^4 + 13x^2 + 36
\][/tex]
Therefore, the polynomial function with a leading coefficient of 1 and roots [tex]\(2i\)[/tex] and [tex]\(3i\)[/tex] each with a multiplicity of 1 is:
[tex]\[
f(x) = x^4 + 13x^2 + 36
\][/tex]
This matches with the option:
[tex]\[
\boxed{(x + 2i)(x + 3i)(x - 2i)(x - 3i)}
\][/tex]
