First, recall the Conjugate Zeros Theorem:
Let f(x) be a polynomial function with real coefficients. If the complex number a + bi is a zero of f(x), then the conjugate of a + bi is also a zero of f(x).
Since we are given that 4i is a zero, we know that -4i is also a zero. The polynomial function of interest has degree 3, so we know that the zeros of f(x) are 4, 4i, and -4i. Now recall the Factor Theorem:
r is a zero of the polynomial function f(x) if and only if x - r is a factor of f(x).
Therefore, we can write f(x) as
f(x) = A(x - 4)(x - 4i)(x + 4i)
We can now find A using the fact that f(-1) = -85
f(-1) = -85
A(-1 - 4)(-1 - 4i)(-1 + 4i) = -85.
Noting that (-1 - 4i)(-1 + 4i) = 17, this becomes
-85A = -85, so
A = 1.
Hence, f(x) = 1(x - 4)(x -4i)(x + 4i). Multiply out the factors on the right side to get the expanded form.
f(x) = x3 - 4x2 + 16x - 64.
Hope that helps! Let me know if you need any further explanation.