William P. answered • 09/05/19

University Math Instructor and Experienced Calculus Tutor

Hello Elly,

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) = x**^{3}** - 4x**^{2}** + 16x - 64**.

Hope that helps! Let me know if you need any further explanation.

William

William P.

09/07/19

Elly S.

Thank you, that has been extremely helpful!09/05/19