The value of (a^2-b^2)^3(b^2-c^2)+(c^2-a^2)^3 (a-b)^3+(b-c)^3+(c-… - Vidyadip

MCQ

The value of $\frac{(\text{a}^2-\text{b}^2)^3(\text{b}^2-\text{c}^2)+(\text{c}^2-\text{a}^2)^3}{(\text{a}-\text{b})^3+(\text{b}-\text{c})^3+(\text{c}-\text{a})^3}$ is:

A
$3(a - b) (b - c) (c - a)$
✓
$3(a + b) (b + c) (c + a)$
C
$3(a + b) (b + c) (c + a) (a - b) (b - c) (c - a)$
D
None of these.

✓

Answer

Correct option: B.

$3(a + b) (b + c) (c + a)$

$\frac{(\text{a}^2-\text{b}^2)^3(\text{b}^2-\text{c}^2)+(\text{c}^2-\text{a}^2)^3}{(\text{a}-\text{b})^3+(\text{b}-\text{c})^3+(\text{c}-\text{a})^3}$
$=\frac{3(\text{a}^2-\text{b}^2)(\text{b}^2-\text{c}^2)(\text{c}^2-\text{a}^2)}{3(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{a})}$ $[\text{Since}\text{ x}^3+\text{y}^3+\text{z}^3=3\text{xyz},\text{ if}\text{ x}+\text{y}+\text{z}=0]$
$=\frac{3(\text{a}-\text{b})(\text{a}+\text{b})(\text{b}-\text{c})(\text{b}+\text{c})(\text{c}-\text{a})(\text{c}+\text{a})}{3(\text{a}-\text{b})(\text{b}-\text{c}(\text{c})-\text{a})}$
$=3(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a})$

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