So expression becomes: x+y+z−3 Step 2: Take LCM of all 3 terms: LCM=(a−b)(b−c)(c−a) Write all terms with common denominator: x=
(a−b)2
(b−c)(c−a)
=
(a−b)3
(a−b)(b−c)(c−a)
y=
(b−c)2
(c−a)(a−b)
=
(b−c)3
(b−c)(c−a)(a−b)
z=
(c−a)2
(a−b)(b−c)
=
(c−a)3
(c−a)(a−b)(b−c)
Numerators: ⇒x+y+z=
(a−b)3+(b−c)3+(c−a)3
(a−b)(b−c)(c−a)
Use identity: If x+y+z=0⇒x3+y3+z3=3xyz Here, let: u=(a−b),v=(b−c),w=(c−a) Then: u+v+w=0 So: u3+v3+w3=3uvw Apply: Numerator =(a−b)3+(b−c)3+(c−a)3=3(a−b)(b−c)(c−a) Denominator =(a−b)(b−c)(c−a) So: x+y+z=
