Question

factorise (a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3​

Answer 1

$We \:know \:that , \\If \: x + y + z = 0 \: then \\x^{3} + y^{3} + z^{3} = 3xyz$

$Here, x = a^{2} - b^{2} , \: y = b^{2} - c^{2} \: and \\z = c^{2} - a^{2}$

$x + y + z \\= a^{2} - b^{2} + b^{2} - c^{2} + c^{2} - a^{2} \\= 0$

$Now, x^{3} + y^{3} + z^{3} \\= ( a^{2} - b^{2} )^{3} +( b^{2} - c^{2})^{3} +( c^{2} - a^{2} )^{3}\\= 3( a^{2} - b^{2} )( b^{2} - c^{2} )(c^{2} - a^{2})$

Therefore.,

$\red {( a^{2} - b^{2} )^{3} +( b^{2} - c^{2})^{3} +( c^{2} - a^{2} )^{3}}$

$\green {= 3( a^{2} - b^{2} )( b^{2} - c^{2} )(c^{2} - a^{2}) }$

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