Answer:
-4x^3 + 20x^2 - 16x - 40.
Step-by-step explanation:
Complex zeroes exist in conjugate pairs so the third root must be 3 - i.
So knowing the 3 zeroes we can write:
(x + 1)(x - (3 - i))(x - (3 + i)) = 0
= (x + 1)(x - 3 + i)(x - 3 - i)
= (x + 1)( x^2 - 3x - ix - 3x + 9 + 3i + ix - 3i - i^2)
= (x + 1)(x^2 - 6x + 10)
= x^3 - 6x^2 + 10x + x^2 - 6x + 10
= x^3 - 5x^2 + 4x + 10.
Now the y -intercept is -40 so substituting x = 0 in the polynomial
y = 10
We have multiply the whole equation by -4). giving
-4x^3 + 20x^2 - 16x - 40. (answer).
Using the Factor Theorem, the function is given by:
[tex]f(x) = -4(x^3 - 5x^2 + 4x + 10)[/tex]
The Factor Theorem states that a polynomial function [tex]f(x)[/tex] with roots [tex]x_1, x_2, ..., x_n[/tex] is written as:
[tex]f(x) = a(x - x_1)(x - x_2)...(x - x_n)[/tex]
In which a is the leading coefficient.
In this problem:
Then:
[tex]f(x) = a(x - x_1)(x - x_2)...(x - x_n)[/tex]
[tex]f(x) = a(x + 1)(x - 3 - i)(x - 3 + i)[/tex]
[tex]f(x) = a(x + 1)(x^2 - 6x + 9 - i^2)[/tex]
[tex]f(x) = a(x + 1)(x^2 - 6x + 10)[/tex]
[tex]f(x) = a(x^3 - 5x^2 + 4x + 10)[/tex]
y-intercept is -40, which means that when [tex]x = 0, f(x) = -40[/tex]. This is used to find a.
[tex]-40 = a(0^3 - 5(0)^2 + 4(0) + 10)[/tex]
[tex]10a = -40[/tex]
[tex]a = -\frac{40}{10}[/tex]
[tex]a = -4[/tex]
Then, the function is:
[tex]f(x) = -4(x^3 - 5x^2 + 4x + 10)[/tex]
A similar problem is given at https://brainly.com/question/24380382
