Question

(b^2-c^2)^3+(c^2-a^2)+(a^2-b^2)^3/(b-c)^3+(c-a)^3+(a-b)^3
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Answer 1

Answer:

On simplifying we get ( a + b )( b + c )( c + a )

Step-by-step explanation:

Given Expression:

\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}(a−b)3+(b−c)3+(c−a)3(a2−b2)3+(b2−c2)3+(c2−a2)3

We need to simplify given expression.

We use the following result,

if x + y + z = 0 then x³ + y³ + z³ = 3xyz

First x = a² - b² , y = b² - a² and z = c² - a²

⇒ x + y + z = a² - b² + b² - c² + c² - a² = 0

So, (a² - b²)³ + (b² - a²)³ + (c² - a²) = 3(a² - b²)(b² - a²)(c² - a²)

Second, x = a - b  ,  y = b - c  and z = c - a

⇒ x + y + z = a - b + b - c + c - a = 0

So,  (a - b)³ + (b - c)³ + (c - a)³ = 3(a - b)(b - c)(c - a)

Thus, we get

\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}(a−b)3+(b−c)3+(c−a)3(a2−b2)3+(b2−c2)3+(c2−a2)3

=\frac{3(a^2-b^2)(b^2-c^2)(c^2-a^2)}{3(a-b)(b-c)(c-a)}=3(a−b)(b−c)(c−a)3(a2−b2)(b2−c2)(c2−a2)

Now using, x² - y² = ( x - y )( x + y )

=\frac{3(a-b)(a+b)(b-c)(b+c)(c-a)(c+a)}{3(a-b)(b-c)(c-a)}=3(a−b)(b−c)(c−a)3(a−b)(a+b)(b−c)(b+c)(c−a)(c+a)

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

Therefore, On simplifying we get ( a + b )( b + c )( c + a )