If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0$ , Find the equation whose roots are $(α - β) (α - β), (β - γ) (β - α), (γ - α) (γ - β)$ .

$x^{3} - 5 x^{2} - 36 = 0$

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$x^{3} - 9 x^{2} - 216 = 0$

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$x^{3} - 11 x^{2} - 100 = 0$

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$x^{3} - 9 x^{2} - 225 = 0$

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Solution

The correct option is B
$x^{3} - 9 x^{2} - 216 = 0$

$α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0$ __(1)
Using the relation between roots and coefficients, we get
$α + β + γ = \frac{- b}{a} = 0 \Rightarrow β + γ = - α$ ,

$α β + β γ + γ α = \frac{c}{a} = 3$ ,
$α γ β = \frac{- d}{a} = - 2 \Rightarrow β γ = \frac{- 2}{α}$

Let $y = (α - β) (α - γ)$

$y = (α - β) (α - γ) = α^{2} - α β - α + β$
$= α^{2} - α (β + γ) + β γ β + γ = - α, β γ = \frac{- 2}{α}$

$= α^{2} - α (- α) + \frac{- 2}{α}$

$\Rightarrow y = 2 α^{2} - \frac{2}{α}$

$\Rightarrow y α = 2 α^{3} - 2$

$\Rightarrow 2 α^{3} - y α - 2 = 0$ _(2)

To generalize this equation,

Replace $α$ by $x$ , we get

2 $x^{3}$ -yx - 2 = 0 ________(3)

To get the relation between x and y

Subtracting equation (3) from twice of equation (1)

2 $x^{3}$ -xy - 2 - 2 $x^{3}$ - 6x - 4 = 0

-xy - 6x - 6 = 0

x(6 + y) = - 6

Now replace

x = $\frac{- 6}{6 + y}$ in equation (1)

- $\frac{216}{{(6 + y)}^{3}}$ - $\frac{18}{6 + y}$ + 2 = 0

${(y + 6)}^{3} - 9 {(y + 6)}^{2} - 108 = 0$

$y^{3} + 9 y^{2} - 216 = 0$

Replace y by x $\Rightarrow x^{3} + 9 x^{3} - 216 = 0$

$x^{3} + 9 x^{2} - 216 = 0$ has roots

$(α - β) (α - γ), (β - γ) (β - α), (γ - α) (γ - β)$

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Similar questions

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α, β, γ

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If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0... (i) and (α - β) (α - γ), (β - γ) (β - α), (γ - α) (γ - β)$ are the roots of equation $y^{3} - 9 y^{2} - 216 = 0 . . . (i i)$ .

Relation to transform the eq.(i) to eq.(ii) is given by .

If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0$ , Find the equation whose roots are $α - β) (α - β), (β - γ) (β - α), (γ - α) (γ - β$ .

If α, β and γ are the roots of the equation $x^{3}$ - 3 $x^{2}$ + 5x - 9 = 0 then the value of the expression

( α + β - γ) ( β + γ -α) (γ + α - β).

Q. Solve the problem

I f α, β, γ a r e t h e r o o t s o f t h e e q u a t i o n x^{3} - 3 x + 11 = 0 t h e n f i n d t h e e q u a t i o n w h o s e r o o t s a r e (α + β), (β + γ) a n d (γ + α) .

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If α,β and γ are the roots of the equation x3+3x+2=0 , Find the equation whose roots are (α−β)(α−β),(β−γ)(β−α),(γ−α)(γ−β).

If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0$ , Find the equation whose roots are $(α - β) (α - β), (β - γ) (β - α), (γ - α) (γ - β)$ .