Answer:
p(x) = [tex]x^{4}[/tex] - x³ + 2x² - 4x - 8
Step-by-step explanation:
Note that complex roots occur in conjugate pairs.
Given x = - 2i is a root then x = 2i is also a root
The roots of the polynomial are x = - 1, x = 2, x = - 2i and x = 2i
Thus the factors of the polynomial are
(x + 1), (x - 2), (x - 2i) and (x + 2i)
The polynomial is the product of it's roots, thus
p(x) = (x + 1)(x - 2)(x - 2i)(x + 2i) ← expand in pairs
= (x² - x - 2)(x² - 4i²) → note i² = - 1, so
= (x² - x - 2)(x² + 4) ← distribute parenthesis
= [tex]x^{4}[/tex] + 4x² - x³ - 4x - 2x² - 8 ← collect like terms
= [tex]x^{4}[/tex] - x³ + 2x² - 4x - 8
