MATH SOLVE

3 months ago

Q:
# 9. Find a polynomial of degree 4 whose coefficients are real numbers and that haszeros -1, 2, and -2i.

Accepted Solution

A:

Answer:p(x) = [tex]x^{4}[/tex] - x³ + 2x² - 4x - 8Step-by-step explanation:Note that complex roots occur in conjugate pairs.Given x = - 2i is a root then x = 2i is also a rootThe roots of the polynomial are x = - 1, x = 2, x = - 2i and x = 2iThus the factors of the polynomial are(x + 1), (x - 2), (x - 2i) and (x + 2i)The polynomial is the product of it's roots, thusp(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