Answer:
[tex]f(x)=x^2+16[/tex]
Step-by-step explanation:
By the conjugate rule, if -4i is a root, then so is +4i. So we have 2 roots, thus, we have a second degree polynomial (namely, a quadratic). If
x = -4i, then
x + 4i is a root.
If
x = 4i, then
x - 4i is a root.
Having (x - 4i)(x + 4i) as roots, we can now FOIL them together to get a polynomial of least degree.
FOILing gives us
[tex]x^2+4ix-4ix-16i^2[/tex]
Notice that the +4ix and the -4ix cancel each other out, leaving you with
[tex]x^2-16i^2[/tex]
Since
[tex]i^2=-1[/tex]
we can make the substitution:
[tex]x^2-16(-1)[/tex]
which simplifies to
[tex]x^2+16[/tex]
In function notation form:
[tex]f(x)=x^2+16[/tex]
