The following are zeros of a polynomial function: - 4, - 3i

1. Identify the third zero of the function.

2. Find the polynomial for which these are zeros. Show your steps.

see explanation

Step-by-step explanation:

(1)

Complex roots occur in conjugate pairs.

Given x = - 3i is a zero, then x = + 3i is also a zero

(2)

Given the zeros x = - 4, x = 3i, x = - 3i, then the factors are

(x + 4), (x - 3i) and (x - ( - 3i)), that is

(x + 4), (x - 3i), (x + 3i) and the polynomial is the product of the factors

f (x) = (x + 4) (x - 3i) (x + 3i) ← expand complex factors

= (x + 4) (x² - 9i²) → i² = - 1, so

= (x + 4) (x² + 9) ← distribute factors

= x³ + 9x + 4x² + 36

= x³ + 4x² + 9x + 36