The expression (a - b)^3 + (b - c)^3 + (c - a)^3 can be factorize… - Vidyadip

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)$

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