Whats all roots of the polynomial P(x) = x ^4 + 9.?

Question

Find all roots of the polynomial P(x) = x^ 4 + 9.

a. (±√ 3 ± i √3)/2

b. (−√ 6 ± i √6)/2

c. ± √3i and ±√ 3

d. (± √6 ± i √6)/2

e. (± √3 ± i √6)/2

Answer 1

Have you learned De Moivre's Theorem?

if z = r (cos(a) + i sin(a) ) then
z^n = r^n(cos(na) + i sin(na) )

-9 = 9 (cos 180 + i sin 180)

r^4 = 9 r = √ 3

4a = 180, a = 45

so the root = √ 3 (cos 45 ± sin 45)
= √ 3 (±√ 2/2 ± √ 2/2
= (± √6 ± i √6)/2

Answer 2

hi

x^4 = -9
x^2 = +/- 3 i
x = √3 √(±i)


√+i = ± √2/2 ( +1 + i )
√-i = ± √2/2 ( +1 - i )

x = √6/2 ( ± 1 ± i)
d)

bye