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

# A) f(x) = x^{3} − 5x^{2} + 20x − 16 B) f(x) = x^{3} − 5x^{2} + 16x − 80

# C) f(x) = x^{3} − 20x^{2} + 5x − 16 D) f(x) = x^{3} − 16x^{2} + 80x − 5

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

## Answer: The function with roots as 5, 4i, -4i is option B, f(x) = x^{3} − 5x^{2} + 16x − 80

Go through the step-by-step solution and deduce the correct answer to the problem.

**Explanation:**

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) (x^{2} + 16)

⇒ f(x) = x^{3} − 5x^{2} + 16x − 80