If α, β and γ are the roots of ax^3 + bx + c = 0 then frame an equation whose roots are α^2 , β^2 , γ^2 .


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Question Number 202436 by MATHEMATICSAM last updated on 26/Dec/23

$If α, β and γ are the roots of$ ${a x}^{3} + b x + c = 0 then frame an equation$ $whose roots are α^{2}, β^{2}, γ^{2} .$
	Answered by aleks041103 last updated on 26/Dec/23

	${a x}^{3} + b x + c = 0, x = α, β, γ$ $\Rightarrow - \frac{c}{a} = α β γ$ $\frac{b}{a} = α β + β γ + α γ$ $0 = α + β + γ$ ${A x}^{3} + {B x}^{2} + C x + D = 0, x = α^{2}, β^{2}, γ^{2}$ $- \frac{D}{A} = α^{2} β^{2} γ^{2} = {(α β γ)}^{2} = \frac{c^{2}}{a^{2}}$ $\frac{C}{A} = α^{2} β^{2} + β^{2} γ^{2} + α^{2} γ^{2} = {(α β + β γ + α γ)}^{2} - 2 α β γ (α + β + γ) =$ $= \frac{b^{2}}{a^{2}} + 2 \frac{c}{a} . 0 = \frac{b^{2}}{a^{2}}$ $- \frac{B}{A} = α^{2} + β^{2} + γ^{2} = {(α + β + γ)}^{2} - 2 (α β + β γ + α γ) =$ $= 0^{2} - 2 \frac{b}{a} = - \frac{2 b}{a}$ $x^{3} + \frac{2 b}{a} x^{2} + \frac{b^{2}}{a^{2}} x - \frac{c^{2}}{a^{2}} = 0$ $\Rightarrow a^{2} x^{3} + 2 {a b x}^{2} + b^{2} x - c^{2} = 0$
	Answered by Frix last updated on 27/Dec/23

	$x^{3} + {p x}^{2} + q x + r = 0$ $(\sqrt{x^{3}} + p x + q \sqrt{x} + r) (\sqrt{x^{3}} - p x + q \sqrt{x} - r) = 0$ $x^{3} - (p^{2} - 2 q) x^{2} - (2 p r - q^{2}) x - r^{2} = 0$ $p = 0 \land q = \frac{b}{a} \land r = \frac{c}{a}$ $x^{3} + \frac{2 {b x}^{2}}{a} + \frac{b^{2} x}{a^{3}} - \frac{c^{2}}{a^{2}} = 0$ $a^{2} x^{2} + 2 {a b x}^{2} + b^{2} x - c^{2} = 0$
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