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If x + y + z = 0, then x (y – z)3 + y (z – x)3 + z (x – y)3 equals

(a) 0 

(b) y + z 

(c) 1 

(d) (z + x)2

1 Answer

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Best answer

(a) 0

Now, x + y + z = 0 

⇒ x = –y – z, y = – x – z, z = – x – y 

∴ x (y – z)3 + y (z – x)3 + z (x – y)3 

= (– y – z) (y – z)3 + (– z – x) (z – x)3 + (– x – y) (x – y)3 

= – (y + z) (y – z)3 – (z + x) (z – x)3 – (x + y) (x – y)3 

= – [(y2 – z2) (y – z)2 + (z2 – x2) (z – x)2 + (x2 – y2) (x – y)2

= – [(y2 – z2) (y2 – 2yz + z2) + (z2 – x2) (z2 – 2xz + x2) + (x2 – y2) (x2 – 2xy + y2)]

= 2yz (y2 – z2) + 2xz (z2 – x2) + 2xy (x2 – y2

= 2 (y3z – yz3 + z3x – x3z + x3y – xy3

= 2 [x3 (y – z) + y3 (z – x) + z3 (x – y)] 

= 2 [– (y – z) (z – x) (x – y) . (x + y + z)] 

= 0.                                               ( x + y + z = 0)

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