If ω³ = 1 and ω ≠ 1, then (a + bω + cω²) / (c + aω + bω²) + (a + bω + cω²) / (b + cω + aω²) = ?

Steps to Solve

1. Understand the Properties of ω

Since ( \omega^3 = 1 ) and ( \omega \neq 1 ), the values of ( \omega ) are: $$ \omega = e^{2\pi i/3} \quad \text{and} \quad \omega^2 = e^{-2\pi i/3} $$ The properties of ( \omega ) include: $$ 1 + \omega + \omega^2 = 0 $$

2. Rewrite Each Fraction

We will simplify each of the fractions in the expression. The first term is: $$ \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} $$

The second term is: $$ \frac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} $$

3. Simplify the Denominators

Using the established property ( 1 + \omega + \omega^2 = 0 ), we can express ( \omega^2 ) as ( -1 - \omega ) and substitute to simplify the denominators.

4. Combine the Terms

The expression to solve becomes: $$ S = \frac{A}{c + a\omega + b\omega^2} + \frac{A}{b + c\omega + a\omega^2} $$ where ( A = a + b\omega + c\omega^2 ).

Since we have ( S ), we will aim to combine these fractions using a common denominator.

5. Use Symmetry

Notice that both fractions share the form of their numerators and by symmetry, add the fractions after finding a common denominator. This leads to: $$ S = \frac{A(c + a\omega + b\omega^2) + A(b + c\omega + a\omega^2)}{(c + a\omega + b\omega^2)(b + c\omega + a\omega^2)} $$

6. Evaluate the Results

Using simplifications and properties of ( \omega ), calculate and combine: This leads you to an expression which evaluates to: $$ S = 2 $$