Question

Show that (a+b)³+(b-c)³+(c-a)³=3(a-b) (b-c) (c-a)​

Answer 1

How do you solve that (a²-b²) ³ + (b²-c²) ³ + (c²-a²) ³/(a-b) ³ + (b-c) ³ + (c-a) ³?

(a2−b2)3+(b2−c2)3+(c2−a2)3(a−b)3+(b−c)3+(c−a)3

Let in numerator

x=(a2−b2),y=(b2−c2),z=(c2−a2)

⟹x+y+z=0(1)

We know, (x+y+z)3=x3+y3+z3+3(x+y+z)(xy+yz+zx)−3xyz

0=x3+y3+z3+3(0)(xy+yz+zx)−3xyz

x3+y3+z3=3xyz(2)

Let in denominator,

p=(a−b),q=(b−c),r=(c−a)

⟹p+q+r=0

⟹

p3+q3+r3=3pqr(3)

(a2−b2)3+(b2−c2)3+(c2−a2)3(a−b)3+(b−c)3+(c−a)3=3xyz3pqr

=(a2−b2)(b2−c2)(c2−a2)(a−b)(b−c)(c−a)

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

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