The polynomial function is:
[tex]\boxed{f(x)=a(x^3+5x^2+16x+80)}[/tex]
In this exercise, we will use two important theorems:
In this case we have the following roots:
[tex]x_{1}=-5 \ and \ x=4i[/tex]
According to the Complex Conjugate Root Theorem:
[tex]x_{3}=-4i \\ \\ Is \ also \ a \ root \ of \ the \ polynomial \ function[/tex]
Hence, we can write the function as:
[tex]f(x)=a(x-(-5))(x-4i)(x+4i) \\ \\ f(x)=a(x+5)(x-4i)(x+4i) \\ \\ \\ By \ distributive \ property: \\ \\ f(x)=a(x+5)(x^2+4xi-4xi-(4i)^2) \\ \\ f(x)=a(x+5)(x^2-16i^2) \\ \\ \\ i=\sqrt{-1} \\ \\ i^2=-1 \\ \\ \\ f(x)=a(x+5)(x^2+16) \\ \\ By \ distributive \ property: \\ \\ \boxed{f(x)=a(x^3+5x^2+16x+80)} \\ \\ For \ any \ real \ leading \ coefficient \ a[/tex]
