To solve the equation [tex]\( \frac{a + b \omega + c \omega^2}{b + c \omega + a \omega^2} = \omega \)[/tex], we start by using the properties of the complex cube roots of unity. Let [tex]\( \omega \)[/tex] be a complex cube root of unity, satisfying [tex]\( \omega^3 = 1 \)[/tex] and [tex]\( 1 + \omega + \omega^2 = 0 \)[/tex].
Given the equation:
[tex]\[ \frac{a + b \omega + c \omega^2}{b + c \omega + a \omega^2} = \omega \][/tex]
We can rewrite this as:
[tex]\[ a + b \omega + c \omega^2 = \omega (b + c \omega + a \omega^2) \][/tex]
Expanding the right-hand side:
[tex]\[ a + b \omega + c \omega^2 = \omega b + \omega^2 c + a \omega^3 \][/tex]
Using [tex]\( \omega^3 = 1 \)[/tex], the equation simplifies to:
[tex]\[ a + b \omega + c \omega^2 = \omega b + \omega^2 c + a \][/tex]
Now, collect like terms:
[tex]\[ a - a + b \omega - \omega b + c \omega^2 - \omega^2 c = 0 \][/tex]
This simplifies to:
[tex]\[ (b \omega - \omega b) + (c \omega^2 - \omega^2 c) = 0 \][/tex]
Since the coefficients of [tex]\( \omega \)[/tex] and [tex]\( \omega^2 \)[/tex] are constants, we can separately equate the coefficients of 1, [tex]\( \omega \)[/tex], and [tex]\( \omega^2 \)[/tex] to zero.
By inspection, we see that:
[tex]\[ a = a \text{ (Already balanced, contributing no new information)} \][/tex]
[tex]\[ b \omega - \omega b = 0 \][/tex]
[tex]\[ c \omega^2 - \omega^2 c = 0 \][/tex]
Both [tex]\( b \omega - \omega b \)[/tex] and [tex]\( c \omega^2 - \omega^2 c \)[/tex] reduce to:
[tex]\[ 0 = 0 \][/tex]
Thus, the key insight is examining the equation earlier formed:
[tex]\[ a + b \omega + c \omega^2 = \omega (b + c \omega + a \omega^2) \][/tex]
Given that [tex]\( \omega \)[/tex] is non-zero, the only solution ensuring both sides match appropriately is:
[tex]\[ a = 0, \][/tex]
[tex]\[ b = 0, \][/tex]
[tex]\[ c = 0. \][/tex]
So, the conclusion is:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 1 \][/tex]
[tex]\[ c = -1 \][/tex]
Using unity properties and with calculations, we should have:
[tex]\[ b = a \][/tex]
[tex]\[ c = -a \][/tex]
After checking the setup of the polynomial:
The solution:
\[ a, b, c \ are: \ 1, 1, -1: (1, 1, -1)
