Question

what is cyclic expression

Answer 1

Cyclic Expressions : 
Some expressions involing 3 variables, say a, b and c, remains same, even if a is replaced by b, b is replaced by c and c is replaced by a simultaneously. Such expression is called cyclic expressions. 

Eg: a(b-c)+b(c-a)+c(a-b) 
Some rules of cyclic expressions... 
If (a-b) is a factor of a cyclic expression, then (b-c) and (c-a) are also factors of it. 
If (a+b) is a factor of a cyclic expression, then (b+c) and (c+a) are also factors of it. 

(1) a2(b - c) + b2(c - a) + c2(a - b) 
     = a2b - a2c + b2c - b2a + c2a - c2b  
     = a2b - a2c + cb2 - ab2 + ac2 - bc2  
     = a2b - a2c - ab2 + ac2 + b2c - bc2  
     = a2(b - c) - a(b2 + c2) + bc(b - c)  
     = a2(b - c) - a(b - c)(b + c) + bc(b - c)  
     = (b - c)(a2 - a(b + c) + bc)  
     = (b - c)(a2 - ab - ac + bc)  
     = (b - c)(a(a - b) - c(a - b))  
     = (b - c)(a - b)(a - c)  
     = -(a - b)(b - c)(c - a) 

Answer 2

an algebraic expression which remain unchanged under cyclical replacement of letter involved eg.a+ b+ c.