Question

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

Answer 1

(x-4i)(x+4i)=x^2+16
(x^2+16)(x-5)=x^3-5x^2+16x-80
Answer is B

Answer 2

If a polynomial has roots 5, 4i and -4i, then it can has such three factors:

1. 
   `x-5;`
2. 
   `x-4i;`
3. 
   `x-(-4i)=x+4i.`

Therefore, the expression for this polynomial is

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

Since

`(x-4i)(x+4i)=x^2-(4i)^2=x^2-16i^2=x^2-16\cdot (-1)=x^2+16,`

then

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

Answer: correct choice is B