Write a polynomial equation with roots 5 and - 9i. X^3-? x^2+? X-?=0

x³ - 5x² + 81x - 405 = 0

Step-by-step explanation:

Complex roots occur in conjugate pairs.

Thus given x = - 9i is a root then x = 9i is also a root

The factors are then (x - 5), (x - 9i) and (x + 9i)

The polynomial is the the product of the roots, that is

f (x) = (x - 5) (x - 9i) (x + 9i) ← expand the complex factors

= (x - 5) (x² - 81i²) → note i² = - 1

= (x - 5) (x² + 81) ← distribute

= x³ + 81x - 5x² - 405, thus

x³ - 5x² + 81x - 405 = 0 ← is the polynomial equation