Final answer:
The question involves solving complex polynomial equations and finding roots of unity. Methods include equating bases of equal powers and finding nth roots of complex numbers.
Step-by-step explanation:
The student has provided a series of equations to be solved: (z−1)³ −1=0, z⁴+2=0, and z⁵ −1=i. Additionally, the equation (z+1)⁵=(1−z)⁵ is requested to be solved.
To solve these equations:
- For the equation (z−1)³ −1=0, we would set (z−1)³ equal to 1 and solve for z by taking the cube root of both sides.
- For the equation z⁴+2=0, we would subtract 2 from both sides and then find the fourth roots of -2.
- For the equation z⁵ −1=i, we would add 1 to both sides and then find the fifth roots of 1+i.
- Lastly, for the equation (z+1)⁵=(1−z)⁵, since the powers are the same, we could potentially set the bases equal to each other: z+1=1-z and solve for z. However, as the bases are complex conjugates, we might consider a more nuanced approach or explore the symmetry in complex numbers.