9514 1404 393
Answer:
f(x) = x^3 -4x^2 +4x -16
Explanation:
If 2i is a zero, then its conjugate, -2i, is also a zero. When p is a zero, then (x-p) is a factor. For the three given zeros, the factors are ...
f(x) = (x -4)(x -2i)(x +2i)
f(x) = (x -4)(x^2 +4)
f(x) = x^3 -4x^2 +4x -16 . . . . . . multiplied out to standard form
Now, f(2) = (2 -4)(2^2 +4) = (-2)(8) = -16. This is the required value, so we're done.*
f(x) = x^3 -4x^2 +4x -16
_____
* If the required value were something else, then we would have to multiply the f(x) we found by some factor to make the value of f(2) come out right. That factor would be ...
k = (desired f(2))/(f(2) for our function so far)
In this case, ...
k = -16/-16 = 1