Answer:
f(x) = [tex]$ x^4 - 2x^3 + 49x^2 - 18x + 360 $[/tex] is the required polynomial.
Step-by-step explanation:
Given the zeroes (roots) of the polynomial are [tex]$ -3i $[/tex] and [tex]$ 2 + 6i $[/tex].
We know that complex roots occur in conjugate pairs.
So, this means that [tex]$ +3i $[/tex] and [tex]$ 2 - 6i $[/tex] would also be the roots of the polynomial.
If [tex]$ \pm 3i $[/tex] are to be the roots of the polynomial then the polynomial should have been: [tex]$ x^2 + 9 = 0$[/tex].
Now, to determine the polynomial for which [tex]$ 2 \pm 6i $[/tex] would be the roots.
Roots of the polynomial are nothing but the values of x (any variable) that would make the polynomial zero.
⇒ [tex]$ x = 2 + 6i \hspace{35mm} x = 2 - 6i $[/tex]
⇒ [tex]$ x - 2 - 6i = 0 \hspace{25mm} x - 2 + 6i = 0 $[/tex]
The required polynomial would be the product of all the above polynomials.
[tex]$ i.e., (x^2 + 9)(x - 2 + 6i)(x - 2 - 6i) = 0 $[/tex]
Multiply this to get the required equation.
⇒[tex]$ (x^2 + 9)(x^2 - 2x + 40) $[/tex]
[tex]$ \implies x^4 - 2x^3 + 40x^2 + 9x^2 - 18x + 360 = 0 $[/tex]
∴ The required polynomial is x⁴ - 2x³ + 49x² - 18x + 360 = 0.
