Find all complex solutions to the polynomial equation z^4 + 16 = 0.
Answer and explanation
Correct answer: C. z = sqrt(2) * (+/- 1 +/- i)
- Step 1: Step 1: z^4 = -16 = 16 cis(pi + 2kpi) for k = 0, 1, 2, 3. Step 2. Step 2: Modulus r = 16^(1/4) = 2. Arguments: theta_k = (pi + 2kpi) / 4 = pi/4, 3pi/4, 5pi/4, 7pi/4. z = 2 (cos(theta) + isin(theta)) = 2 (+/- 1/sqrt(2) +/- i/sqrt(2)) = sqrt(2) (+/- 1 +/- i). *Answer: z = sqrt(2) (+/- 1 +/- i). Why not the others:** The solutions form a square rotated by 45 degrees in the complex plane.
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