Question

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.

Answer 1

see explanation

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