a+b+c=1 a^2 +b^2 +c^2 =1 ⇒ abc=? a^3 +b^3 +c^3 =1


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Question Number 147076 by mathdanisur last updated on 17/Jul/21

$a + b + c = 1$ $a^{2} + b^{2} + c^{2} = 1 \Rightarrow a b c = ?$ $a^{3} + b^{3} + c^{3} = 1$
	Answered by gsk2684 last updated on 17/Jul/21

	${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$ $1 = 1 + 2 (a b + b c + c a)$ $a b + b c + c a = 0$ $a n d$ $a^{3} + b^{3} + c^{3} = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) + 3 a b c$ $1 = (1) (1 - 0) + 3 a b c$ $a b c = 0$
	Commented by mathdanisur last updated on 18/Jul/21

	$t h a n k y o u S e r$
	Answered by mr W last updated on 17/Jul/21

	$e_{1} = p_{1} = 1$ $p_{2} = 1 = e_{1}^{2} - 2 e_{2} = 1^{2} - 2 e_{2} \Rightarrow e_{2} = 0$ $p_{3} = 1 = e_{1}^{3} - 3 e_{1} e_{2} + 3 e_{3} = 1^{3} - 3 \times 1 \times 0 + 3 e_{3}$ $\Rightarrow e_{3} = a b c = 0$
	Commented by mathdanisur last updated on 18/Jul/21

	$t h a n k y o u S e r$
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