Question

\((x-y)^3+(y-z)^3+(z-x)^3=\)

• A. \(3xyz\)
• B. \((x-y)(y-z)(z-x)\)
• C. \(3(x-y)(y-z)(z-x)\)
• D. 0

Hint:

Whenever three expressions add up to zero, check whether the identity \(a^3+b^3+c^3=3abc\) can be applied.

Answer 1

The Correct Option is C

Concept: A standard identity states that if \[ a+b+c=0 \] then \[ a^3+b^3+c^3=3abc \] We shall transform the given expression into this form.

Step 1: Choose suitable substitutions.
Let \[ a=x-y,\qquad b=y-z,\qquad c=z-x \] Then \[ a+b+c = (x-y)+(y-z)+(z-x) \] \[ =x-y+y-z+z-x \] \[ =0 \] Thus the condition for the identity is satisfied.

Step 2: Apply the identity.
Since \(a+b+c=0\), \[ a^3+b^3+c^3=3abc \] Substituting back, \[ (x-y)^3+(y-z)^3+(z-x)^3 = 3(x-y)(y-z)(z-x) \] \(3(x-y)(y-z)(z-x)\)