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