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Definition of Functions

If α, β, γ ...

Question

If $α, β, γ$ are the roots of the equation $x^{3} - 3 x + 11 = 0$ , what is the equation whose roots are $(α + β), (β + γ)$ and $(γ + α)$ ?

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Solution

given equation $x^{3} - 3 x + 11 = 0$
$α + β + γ = 0$ $α, β, γ$ are roots
$α β + β γ + γ α = - 3$ of the equation
$α β γ = - 11$
if $α + β, β + γ, γ + α$ are roots then
Equation say $a x^{3} + b x^{2} + c x + d = 0$
$(α + β) + (β + γ) + (γ + α) = - b / a .$
$2 (α + β + γ) = - b / a \Rightarrow b / a = 0$
$(α + β) (β + γ) + (β + γ) (γ + α) + (γ + α) (α + β) = c / a$
We know that $α + β + γ = 0$
$∴ α + β = - γ$ $β + γ = - α$ $γ + α = - β$
$(- γ \times - α) + (- β \times - α) + (- β \times - γ) = c / a$
$α β + β γ + γ α = c / a$
$\Rightarrow c / a = - 3$
$(α + β) (β + γ) (γ + α) = - d / a$
$- α β γ = - d / a$
$\Rightarrow d / a = - 11$
$a x^{3} + b x^{2} + c x + d = 0 \Rightarrow x^{3} + \frac{b}{a} x^{2} + \frac{c}{a} x^{2} + \frac{c}{a} x + \frac{d}{a} = 0$
Req. Equation is $x^{3} - 3 x - 11 = 0$

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