Which of the following is a polynomial with roots 5, 4i, and -4i?
A) f(x) = x³ - 5x² + 20x - 16
B) f(x) = x³ - 5x² + 16x - 80 
C) f(x) = x³ - 20x² + 5x - 16
D) f(x) = x³ - 16x² + 80x - 5

Solution:

A polynomial can have both real as well as imaginary roots in it.

Let us suppose the zeros of the polynomial be x = a, x = b, and x = c

Therefore, the factors can be written as, (x - a), (x - b), (x - c), and the polynomial is the product of the factors.

f(x) = k(x - a)(x - b)(x - c), where k is a multiplier.

Here, x = 5, x = 4i, x = -4i are the factors of the polynomial.

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

Let k = 1.

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

f(x) =  (x - 5) (x² + 16)

⇒ f(x) = x³ − 5x² + 16x − 80

Therefore, the function with roots as 5, 4i, -4i is option B, f(x) = x³ - 5x² + 16x - 80.

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

Summary:

The function with roots as 5, 4i, -4i is option B, f(x) = x³ - 5x² + 16x - 80