If f(x)=a+b x+c x2 and α, β, γ are roots of the equation x3=1, then |a&b&c b&c&a c&a&b| is equal to

Question

If f(x)=a+b x+c x2 and α, β, &gamma; are roots of the equation x3=1, then |a&b&c b&c&a c&a&b| is equal to (A) f(α)+f(β)+f(&gamma;) (B) f(α) f(β)+f(β) f(&gamma;)+f(&gamma;) f(α) (C) f(α) f(β) f(&gamma;) (D) -f(α) f(β) f(&gamma;)

Tardigrade Admin · Accepted Answer

|a&b&c b&c&a c&a&b| =-(a3+b3+c3-3abc) =-(a+b+c)(a+b ω2+c ω)(a+b ω+c ω2) =-f(α) f(β) f(&gamma;) [∵ α=1, β=ω, &gamma;=ω2]

Q. If $f (x) = a + b x + c x^{2}$ and $α, β, γ$ are roots of the equation $x^{3} = 1$ , then $∣ ∣ a b c b c a c a b ∣ ∣$ is equal to

$f (α) + f (β) + f (γ)$

$f (α) f (β) + f (β) f (γ) + f (γ) f (α)$

$f (α) f (β) f (γ)$

$- f (α) f (β) f (γ)$

$∣ ∣ a b c b c a c a b ∣ ∣ = - (a^{3} + b^{3} + c^{3} - 3 ab c)$
$= - (a + b + c) (a + b ω^{2} + c ω) (a + bω + c ω^{2})$
$= - f (α) f (β) f (γ)$
$[∵ α = 1, β = ω, γ = ω^{2}]$

Q. If f(x)=a+bx+cx2 and α,β,γ are roots of the equation x3=1, then ∣∣​abc​bca​cab​∣∣​ is equal to

f(α)+f(β)+f(γ)

f(α)f(β)+f(β)f(γ)+f(γ)f(α)

f(α)f(β)f(γ)

−f(α)f(β)f(γ)