Question

Find the value of (x-y)³+(y-z)³+(z-x)³​

Answer 1

Main concept:A^3 +B^3+ C^3=3ABC

if A+B+C=O

Given A=(X-Y)

B=(Y-Z)

C=(Z-X)

Their sum is A +B+C=O

so, their value is 3ABC

=3×(X-Y)(Y-Z)(Z-X)

Solve it from here.

Answer 2

The value of (x-y)³+(y-z)³+(z-x)³​ = 3x²(z-y) + 3y²(x-z) + 3z²(y-x)

Given:

(x-y)³+(y-z)³+(z-x)³​

To find:

Find the value of (x-y)³+(y-z)³+(z-x)³​

Solution:

From algebraic identities

(a -b)³ = a³- 3a²b + 3ab²- b³

From the above formula

(x - y)³ = x³- 3x²y + 3xy²- y³

(y - z)³ = y³- 3y²z + 3yz²- z³

(z - x)³ ​= z³- 3z²x + 3zx²- x³

⇒ (x-y)³+(y-z)³+(z-x)³​

= x³- 3x²y + 3xy²- y³ +  y³- 3y²z + 3yz²- z³ + z³- 3z²x + 3zx²- x³  

= -3x²y + 3xy² - 3y²z + 3yz² - 3z²x + 3zx²

=  3x²(z-y) + 3y²(x-z) + 3z²(y-x)

The value of (x-y)³+(y-z)³+(z-x)³​ = 3x²(z-y) + 3y²(x-z) + 3z²(y-x)  

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