MCQ
The expression $(a - b)^3 + (b - c)^3 + (c - a)^3$ can be factorized as:
- A$(a - b)(b - c)(c - a)$
- ✓$3(a - b)(b - c)(c - a)$
- C$-3(a - b)(b - c)(c - a)$
- D$(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
✓
Answer
Correct option: B.
$3(a - b)(b - c)(c - a)$
By we know that $a^3 + b^3 + c^3 - 3abc$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
If $a + b + c = 0$, then
$a^3 + b^3 + c^3 = 3abc$
In given expression,
Let $a - b = A, b - c = B, c - a = C$
Now, $a - b + b - c + c - a = 0$
i.e.$ A + B + C = 0$
$\Rightarrow A^3 + B^3 + C^3 = 3\text{ABC}$
$\Rightarrow (a - b)^3 + (b - c)^3 + (c - a)^3 $
$= 3(a - b)(b - c)(c - a)$
Hence, correct option is $(b).$
$= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
If $a + b + c = 0$, then
$a^3 + b^3 + c^3 = 3abc$
In given expression,
Let $a - b = A, b - c = B, c - a = C$
Now, $a - b + b - c + c - a = 0$
i.e.$ A + B + C = 0$
$\Rightarrow A^3 + B^3 + C^3 = 3\text{ABC}$
$\Rightarrow (a - b)^3 + (b - c)^3 + (c - a)^3 $
$= 3(a - b)(b - c)(c - a)$
Hence, correct option is $(b).$
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