Mitkä ovat seuraavan monimutkaisen polynomin neljä juuria: P (z) = z ^ 4 - 2z ^ 2 + 4

Vastaus:

#z = pm sqrt6 / 2 pm i sqrt (2) / 2 #

Selitys:

# z ^ 4 - 2 * z ^ 2 + 4 = 0 #

#Delta = 4 - 4 * 1 * 4 = -12 #

# z ^ 2 = (2 pm 2 i sqrt 3) / 2 #

# z ^ 2 = 2 (1/2 pm i sqrt 3/2) #

# z ^ 2 = 2 (cos frac {pi} {3} pm i sin frac {pi} {3}) #

#z = pm sqrt2 (cos frac {pi} {6} pm i sin frac {pi} {6}) #

#z = pm sqrt2 (sqrt3 / 2 pm i / 2) #

#z = pm sqrt6 / 2 pm i sqrt (2) / 2 #