Question

simplify (a2-b2)3+(b2-c2)3+(c2-a2)3divided by (a-b)3+(b-c)3+(c-a)3

Answer 1

We know that if a+b+c+0
Then x³+y³+c³=3xyz
Also in this question
Numerator if x=a²-b² y=b²-c² z=c²-a²
a²-b²+b²-c²+c²-a²=0
So numerator =3(a²-b²)(b²-c²)(c²-a²)
Now denominator
If x=a-b y=b-c z=c-a
Here also x+y+z=0
So denominator =3(a-b)(b-c)(c-a)
Hence the answer is
(a+b) (b+c) (c+a)

Answer 2

Answer:

We know,

if a + b + c = 0 ; a³ + b³ + c³ = 3abc

We observe,

( a² - b² ) + ( b² - c² ) + ( c² - a² ) = 0

=> ( a² - b² )³ + ( b² - c² )³ + ( c² - a² )³ = 3( a² - b² )( b² - c² )( c² - a² )

Similarly, 

( a - b ) + ( b - c ) + ( c - a ) = 0

=> ( a - b )³ + ( b - c )³ + ( c - a )³ = 3( a - b )( b - c )( c - a )

Then , [ ( a² - b² )³ + ( b² - c² )³ + ( c² - a² )³ ] / [ ( a - b )³ + ( b - c )³ + ( c - a )³ ]

=3( a² - b² )( b² - c² )( c² - a² ) / 3( a - b )( b - c )( c - a )

= ( a + b )( b + c )( c + a )