Part 4/5:
Similarly, we compute ( y_3 ):
[
y_3 = \frac{z_3 - p}{3z_3}
]
Replacing ( z_3 ) with ( z_1 \times W_2 ):
[
y_3 = \frac{(z_1 \times W_2) - p}{3(z_1 \times W_2)}
]
This equation mirrors the complexity of ( y_2 ), demonstrating how the intricate relationships between cube roots translate back into the quotient required for ( y ).
Conclusion
Through the methodical breakdown of cube roots and their application to complex equations, we have successfully modeled the relationships between ( y ) and its corresponding ( z ) values guided by cube roots of unity. Each derived solution illustrates the elegance of algebraic manipulation and the beauty inherent in solving cubic equations.