To find a polynomial function with rational coefficients that has given zeros, we need to include their conjugates as well. The given zeros are [tex]\( \sqrt{2} \)[/tex] and [tex]\( 3i \)[/tex]. Their conjugates are [tex]\( -\sqrt{2} \)[/tex] and [tex]\( -3i \)[/tex], respectively.
Thus, the zeros of the polynomial are:
- [tex]\( \sqrt{2} \)[/tex]
- [tex]\( -\sqrt{2} \)[/tex]
- [tex]\( 3i \)[/tex]
- [tex]\( -3i \)[/tex]
A polynomial with these roots can be constructed by forming factors from the roots and multiplying them together. The factors of the polynomial are [tex]\( (x - \text{root}) \)[/tex].
Using the roots:
1. For [tex]\( \sqrt{2} \)[/tex] and [tex]\( -\sqrt{2} \)[/tex]:
The polynomial factor is:
[tex]\[
(x - \sqrt{2})(x + \sqrt{2})
\][/tex]
Multiply these factors:
[tex]\[
(x - \sqrt{2})(x + \sqrt{2}) = x^2 - (\sqrt{2})^2 = x^2 - 2
\][/tex]
2. For [tex]\( 3i \)[/tex] and [tex]\( -3i \)[/tex]:
The polynomial factor is:
[tex]\[
(x - 3i)(x + 3i)
\][/tex]
Multiply these factors:
[tex]\[
(x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9
\][/tex]
Now, multiply the two resulting polynomials together to get the final polynomial:
[tex]\[
(x^2 - 2)(x^2 + 9)
\][/tex]
Expand this product:
[tex]\[
(x^2 - 2)(x^2 + 9) = x^2(x^2 + 9) - 2(x^2 + 9)
\][/tex]
[tex]\[
= x^4 + 9x^2 - 2x^2 - 18
\][/tex]
[tex]\[
= x^4 + 7x^2 - 18
\][/tex]
Thus, the polynomial function in expanded form is:
[tex]\[
f(x) = x^4 + 7x^2 - 18
\][/tex]
