Question

factorize the following: (a-b)^3+(b-c)^3+(c-a)^3

Answer 1

Answer:

3a² + 3b² + 3c² - 3ab - 3bc - 3ac

Step-by-step explanation:

(a - b)³ + (b - c)³ + (c - a)³

Let (a - b)³ be A

Let (b - c)³ be B

Let (c - a)³ be C

∴ A³ + B³ + C³

According to the Identity of a³ + b³ + c³ - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

a³ + b³ + c³ = 3abc

∴ A³ + B³ + C³ = 3ABC

= 3 × (a - b)(b - c)(c - a)

= 3 × [(a - b)(b - c)] - [(b - c)(c - a)] - [(c - a)(a - b)]

= 3 × [ab - b² - ac + bc - bc  + c² - ab + ac + ca + a² - bc - ab]

= 3 × (a² + b² + c² - ab - bc - ac)

= 3a² + 3b² + 3c² - 3ab - 3bc - 3ac

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