if 3 X + Y + Z = 0 show that 27 x cube + y cube + Z cube equals 9 x y z
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Step-by-step explanation:
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x Using the identity and proof: x
3
+y
3
+z
3
−3xyz=(x+y+z)(x
2
+y
2
+z
2
−xy−yz−zx).
3
+y
3
+z
3
−3xyz=(x+y+z)(x
2
+y
2
+z
2
−xy−yz−zx)
⇒x
3
+y
3
+z
3
−3xyz=0×(x
2
+y
2
+z
2
−xy−yz−zx)
⇒x
3
+y
3
+z
3
−3xyz=0
Now, given that
If 3x+y+z=0
then,
(3x)
3
+y
3
+z
3
−3×3xyz=(3x+y+z)((3x)
2
+y
2
+z
2
−xy−yz−zx)
⇒27x
3
+y
3
+z
3
−9xyz=0×(9x
2
+y
2
+z
2
−xy−yz−zx)
⇒x
3
+y
3
+z
3
=9xyz
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