Answer:
f(x) = x^4 + 20x^2 +64
Step-by-step explanation:
Complex roots come in conjugate pairs
If we have a root a+bi, we must have a root a-bi
We have a root 2i so we must have a root -2i
We have a root -4i so we must have a root +4i
We have 4 roots 2i,-2i, 4i,-4i
These are all the roots since it is a 4th degree polynomial
Using the zero product property
f(x) = (x-2i) (x - -2i) (x-4i) (x--4i) =0
f(x) = (x-2i) (x +2i) (x-4i) (x+4i)
Multiplying together
= (x^2 +4) ( x^2 + 16)
= x^4 + 4x^2 + 16x^2 + 64
Combining like terms
= x^4 + 20x^2 +64
