Using the Factor Theorem, it is found that the polynomial function is given by:
- [tex]f(x) = a(x^4 - 3x^3 - x^2 - 27x - 90)[/tex], in which a is the leading coefficient.
Factor Theorem:
- The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
- In which a is the leading coefficient.
In this problem, we have that there is a:
- Zero at x = -2, hence [tex]x_1 = -2[/tex].
- Zero at x = 5, hence [tex]x_2 = 5[/tex].
- Zero at x = -3i, hence there also has to be a zero at it's conjugate x = 3i, hence [tex]x_3 = -3i, x_4 = 3i[/tex].
Then, the function is:
[tex]f(x) = a(x + 2)(x - 5)(x + 3i)(x - 3i)[/tex]
[tex]f(x) = a(x^2 - 3x - 10)(x^2 + 9)[/tex]
[tex]f(x) = a(x^4 - 3x^3 - x^2 - 27x - 90)[/tex]
To learn more about the Factor Theorem, you can take a look at https://brainly.com/question/24380382
