Question

The expression \((a-b)^3 + (b-c)^3 + (c-a)^3\) can be factorized as:

• A. \((a - b)(b - c)(c - a)\)
• B. \(3(a - b)(b - c)(c - a)\)
• C. \(3(a + b)(b - c)(c + a)\)
• D. None of the option is correct

Answer 1

The Correct Option is B

Step 1: Let \(x=a-b\), \(y=b-c\), \(z=c-a\), so \(x+y+z=0\).
Step 2: For any three numbers with \(x+y+z=0\), the identity \(x^{3}+y^{3}+z^{3}=3xyz\) holds.
Step 3: Substituting back, \((a-b)^{3}+(b-c)^{3}+(c-a)^{3}=3(a-b)(b-c)(c-a)\).
\[ \boxed{3(a-b)(b-c)(c-a)} \]