Question

prove that; (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)

plzzzzz its urgent. wrong answers will be reported right ans will be marked as brainliest​

Answer 1

Solution:

To prove: (a²-b²)³ + (b²-c²)³ + (c²-a²)³ = 3(a+b)(b+c)(c+a)(a-b)(b-c)(c-a)

Taking L.H.S.,

= (a²-b²)³ + (b²-c²)³ + (c²-a²)³

w.k.t., (x-y)³ = x³-y³-3x²y+3xy² ⇒ (x²-y²)³ = x⁶-y⁶-3x⁴y²+3x²y⁴}

⇒ a⁶-b⁶-3a⁴b²+3a²b⁴+b⁶-c⁶-3b⁴c²+3b²c⁴+c⁶-a⁶-3c⁴a²+3c²a⁴

= 3 [-a⁴b² + a²b⁴ - b⁴c² + b²c⁴ - c⁴a² + c²a⁴]

= 3 [a²b⁴ - b⁴c² + b²c⁴ - a⁴b² - c⁴a² + c²a⁴]

= 3 [-b⁴(c²-a²) + b²(c⁴ - a⁴) - c²a²(c²-a²)]

= 3 [-b⁴(c²-a²) + b²(c²-a²)(c²+a²) - c²a²(c²-a²)]

= 3(c²-a²) [-b⁴ + b²(c²+a²) - c²a²]

= 3(c-a)(c+a) [-b⁴ + b²c² + b²a² - c²a²]

= 3(c+a)(c-a) [-b²(b²-c²) + a²(b²-c²)]

= 3(c+a)(c-a)(b²-c²)(-b² + a²)

= 3(c+a)(c-a)(b+c)(b-c)(a²-b²)

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

= 3(a+b)(b+c)(c+a)(a-b)(b-c)(c-a) = R.H.S.

Hence Proved

Please mark it as Brainliest.