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Math Expert V
Joined: 02 Sep 2009
Posts: 60480
If xyz ≠ 0, is x^3 + y^3 + z^3 = 0?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 42% (01:09) correct 58% (01:46) wrong based on 19 sessions

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If $$xyz ≠ 0$$, is $$x^3 + y^3 + z^3 = 0$$?

(1) $$x^2 + y^2 + z^2=xy + yz + xz$$
(2) $$x + y + z = 0$$

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Math Expert V
Joined: 02 Aug 2009
Posts: 8346
Re: If xyz ≠ 0, is x^3 + y^3 + z^3 = 0?  [#permalink]

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If $$xyz ≠ 0$$, is $$x^3 + y^3 + z^3 = 0$$?

You should know the formula..$$x^3 + y^3 + z^3 -3xyz=(x+y+z)(x^2 + y^2 + z^2-(xy + yz + xz))$$

(1) $$x^2 + y^2 + z^2=xy + yz + xz$$
$$x^2 + y^2 + z^2=xy + yz + xz.......$$$$x^2 + y^2 + z^2-(xy + yz + xz)=0$$...
$$x^3 + y^3 + z^3 -3xyz=(x+y+z)(x^2 + y^2 + z^2-(xy + yz + xz))=(x+y+z)*0=0.....x^3+y^3+z^3=3xyz\neq{0}$$

(2) $$x + y + z = 0$$
$$x^3 + y^3 + z^3 -3xyz=(x+y+z)(x^2 + y^2 + z^2-(xy + yz + xz))=(0)(x^2 + y^2 + z^2-(xy + yz + xz))=0.....x^3+y^3+z^3=3xyz\neq{0}$$

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_________________ Re: If xyz ≠ 0, is x^3 + y^3 + z^3 = 0?   [#permalink] 11 Nov 2019, 08:55
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# If xyz ≠ 0, is x^3 + y^3 + z^3 = 0?  