Final answer:
The equation z⁶ - 1 = 0 factors into (z - 1)(z² + z + 1)(z + 1)(z² - z + 1) = 0. Solving each factor yields two real solutions, z = 1 and z = -1, and four complex solutions from the quadratic factors using the quadratic formula.
Step-by-step explanation:
The equation z⁶ - 1 = 0 can be solved by recognizing it as a difference of squares where z³ is squared. This equation factors into (z³ - 1)(z³ + 1) = 0, which further factors into (z - 1)(z² + z + 1)(z + 1)(z² - z + 1) = 0, using the sum and difference of cubes formulas.
Solving each factor for z gives us:
- z = 1
- z² + z + 1 = 0
- z = -1
- z² - z + 1 = 0
For the quadratic factors, we can use the quadratic formula to find complex solutions: z = (-b ± √(b² - 4ac)) / (2a), leading to two complex solutions for each quadratic factor.
Therefore, the solutions for z⁶ - 1 = 0 are z = 1, z = -1, and four complex numbers which are roots of the quadratic factors.