Question
The value of $(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3$ is:
✓
Answer
$(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3$
Here,
$a^2 - b^2 + b^2 - c^2 + c^2 - a^2 = 0$
Therefore,
$(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 = 3(a^2 - b^2) (b^2 - c^2) (c^2 - a^2)$
$[$Since $x^3 + y^3 + z^3 = 3xyz$ if $x + y + z =0]$
$(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 =$
$3(a + b) (b + c) (c + a) (a - b) (b - c) (c - a)$
Here,
$a^2 - b^2 + b^2 - c^2 + c^2 - a^2 = 0$
Therefore,
$(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 = 3(a^2 - b^2) (b^2 - c^2) (c^2 - a^2)$
$[$Since $x^3 + y^3 + z^3 = 3xyz$ if $x + y + z =0]$
$(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 =$
$3(a + b) (b + c) (c + a) (a - b) (b - c) (c - a)$
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