Find all complex roots of the polynomial equation z^3 + 8 = 0.
Answer and explanation
Correct answer: B. z = -2, 1 + isqrt(3), and 1 - isqrt(3)
- Step 1: z^3 = -8 = 8e^(ipi). Cube roots: z_k = 2e^(i(pi + 2kpi)/3) for k = 0, 1, 2. Step 2. z_0 = 2e^(ipi/3) = 1 + isqrt(3). z_1 = 2e^(ipi) = -2. z_2 = 2e^(i5pi/3) = 1 - isqrt(3). *Answer: z = -2, 1 + isqrt(3), and 1 - isqrt(3). Why not the others:* Sum of roots is 0 and product is -8.
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