First slide

Algebra of complex numbers

Question

If $α, β, γ$ are the cube roots of $p, p < 0,$ then for any x, y and z which does not make denominator zero,
the expression $\frac{x α + y β + z γ}{x β + y γ + z α}$

Moderate

A

$ω, 1$
B

$ω, ω^{2}$
C

$ω^{2}, 1$
D

$1, ω, ω^{2}$

Solution

$x^{3} = p = {(p^{\frac{1}{3}})}^{3} \Rightarrow x = p^{\frac{1}{3}}, p^{\frac{1}{3}} ω, p^{\frac{1}{3}} ω^{2}$
Let $α = p^{\frac{1}{3}}, β = p^{\frac{1}{3}} ω, γ = p^{\frac{1}{3}} ω^{2}$
$\frac{x α + y β + z γ}{x β + y γ + z α} = \frac{x + y ω + ω^{2} z}{x ω + y ω^{2} + z} = \frac{1}{ω} = ω^{2}$
If $α = p^{\frac{1}{3}} ω, β = p^{\frac{1}{3}}, γ = p^{\frac{1}{3}} ω^{2}$ then ,
$\frac{x α + y β + z γ}{x β + y γ + z α} = \frac{1}{ω^{2}} = ω .$

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