Respuesta :
Answer:
f(x) = 5x^3 - 9x^2 + 245x - 441.
Step-by-step explanation:
Complex zeros are in conjugate pairs so the third zero is -7i.
In factor form we have:
f(x) = (x - 7i)(x + 7i)(5x - 9)
= (x^2 - 49i)^2)(5x - 9)
= (x^2 + 49)(5x - 9)
= 5x^3 - 9x^2 + 245x - 441.
Using the factor theorem, the polynomial is given by:
[tex]f(x) = 5x^3 - 9x^2 + 245x - 441[/tex]
The Factor Theorem states that a polynomial function with roots is given by:
[tex]f(x) = a(x - x_1)(x - x_2)...(x - x_n)[/tex]
In which a is the leading coefficient.
In this problem:
- 7i is a zero, thus, it's conjugate -7 also will be, thus [tex]x_1 = 7i, x_2 = -7i[/tex].
- The other zero is [tex]x_3 = \frac{9}{5}[/tex]
Then, considering a leading coefficient of 1:
[tex]f(x) = (x - 7i)(x + 7i)(x - \frac{9}{5})[/tex]
[tex]f(x) = (x^2 + 49)(x - \frac{9}{5})[/tex]
[tex]f(x) = x^3 - \frac{9}{5}x^2 + 49x - \frac{441}{5}[/tex]
Multiplying by 5:
[tex]f(x) = 5x^3 - 9x^2 + 245x - 441[/tex]
A similar problem is given at https://brainly.com/question/11556838
