Answer:
The polynomial is:
[tex]p(x) = x^3 + 2x^2 - 3x + 20[/tex]
Step-by-step explanation:
A third degree polynomial can be written in function of it's zeros [tex]x_1, x_2, x_3[/tex] the following way:
[tex]p(x) = a(x - x_1)(x - x_2)(x - x_3)[/tex]
In which a is the leading coefficient.
Integer coefficient that have zeros: 1+2i, 1-2i, -4
Leading coefficient: 1
So
[tex]p(x) = 1(x - (1+2i))(x - (1-2i))(x - (-4))[/tex]
[tex]p(x) = (x - 1 -2i)(x - 1 + 2i)(x + 4)[/tex]
[tex]p(x) = ((x-1)^2 - (2i)^2)(x + 4)[/tex]
[tex]p(x) = (x^2 - 2x + 1 - 4i^2)(x + 4)[/tex]
Since [tex]i^2 = -1[/tex]
[tex]p(x) = (x^2 - 2x + 1 + 4)(x + 4)[/tex]
[tex]p(x) = (x^2 - 2x + 5)(x + 4)[/tex]
[tex]p(x) = x^3 + 4x^2 - 2x^2 - 8x + 5x + 20[/tex]
[tex]p(x) = x^3 + 2x^2 - 3x + 20[/tex]
