MCQ
$(a - b)^3 + (b - c)^3 + (c - a)^3 =$
- A$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
- B$(a - b)(b - c)(c - a)$
- ✓$3(a - b)(b - c)(c - a)$
- DNone of these.
Let
$a - b = A$
$b - c = B$
$c - a = C$
Now $(A + B + C)^3 = A^3 + B^3 + C^3 + 3(A + B)(B + C)(C + A)$
$⇒ A^3 + B^3 + C^3 = (A + B + C)^3- 3(A + B)(B + C)(C + A)$
Now putting values of $A, B$ and $C.$ we get
$(\text{a} - \text{b})^3 + (\text{b} - \text{c})^3 + (\text{c} - \text{a})^3\\=(\not\text{a}-\not\text{b}+\not\text{b}-\not\text{c}+\not\text{c}-\not\text{a})^3\\-3(\text{a}-\not\text{b}+\not\text{b}-\text{c})(\text{b}-\not\text{c}+\not\text{c}-\text{a})(\text{c}-\not\text{a}+\not\text{a}-\text{b})$
$⇒ (a - b)^3+ (b - c)^3 + (c - a)^3 = 0 - 3 (a - c)(b - a)(c - b)$
$⇒ (a - b)^3+ (b - c)^3 + (c - a)^3 = 3(a - b)(b - c)(c - a)$
Hence, correct option is $(c).$
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