Answer:
[tex]x^{4}[/tex] + 3x² - 4
Step-by-step explanation:
Note that complex zeros occur in conjugate pairs
If 2i is a zero then - 2i is a zero
The zeros are x = 1, x = - 1, x = 2i, x = - 2i, thus the factors are
(x - 1), (x + 1), (x - 2i) and (x + 2i)
The polynomial is expressed as the product of the factors, thus
f(x) = (x - 1)(x + 1)(x - 2i)(x + 2i) ← expanding in pairs
= (x² - 1)(x² - 4i²) → i² = - 1
= (x² - 1)(x² + 4) ← distribute
= [tex]x^{4}[/tex] + 4x² - x² - 4
= [tex]x^{4}[/tex] + 3x² - 4
