Question

Prove that
(y +z - x)3 + (z + x -y)2 + (x + y -z)3

=-24xyz, if x + y + z = 0.
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Answer 1

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Answer 2

Answer:

(y +z - x)³ + (z + x -y)³ + (x + y -z)³ = -24xyz, if x + y + z = 0.

Step-by-step explanation:

Correct Question : prove that

(y +z - x)³ + (z + x -y)³ + (x + y -z)³ = -24xyz, if x + y + z = 0.

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LHS = (y +z - x)³ + (z + x -y)³ + (x + y -z)³

x + y + z = 0  =>

y +z  = -x

​z + x = -y

x + y = -z

putting these values

LHS

= (-x -x)³ + (-y - y)³ + (-z -z)³

= (-2x)³ + (-2y)³ +(-2z)³

= -8x³ - 8y³ - 8z³

= -8 (x³ + y³ + z³)

Now  x + y + z = 0

=> (x + y) = -z

Cubing both sides

x³ + y³ + 3xy(x + y) = (-z)³

=> x³ + y³ + 3xy(-z)  = -z³

=>x³ + y³ + z³ = 3xyz

putting this in LHS

LHS

= -8(3xyz)

= -24xyz

= RHS

QED

Proved

(y +z - x)³ + (z + x -y)³ + (x + y -z)³ = -24xyz