Question

Which of the following is a polynomial with roots 5, 4i, and −4i?

Answer 1

fx= x^3 - 5x^2 + 16x - 80

Answer 2

In order to figure this out, we need to make roots into binomials:

x = 5, 4i, and -4i

This can be written as:

f(x) = (x - 5)(x + 4i)(x - 4i)

Let's take care of the imaginary numbers first:

(x + 4i)(x - 4i)
x^2 -4ix + 4ix + 16
x^2 + 16

Now we can re-insert this into our problem and solve accordingly:

f(x) = (x - 5)(x^2 + 16)

f(x) = x^3 + 16x - 5x^2 - 80

Now we can rearrange the terms in descending order to obtain our polynomial:

f(x) = x^3 - 5x^2 + 16x - 80