2.16 Cyclic Symmetry:
Observe
the expression a+b+c à(1) In the
above expression, suppose we change a to b, b to c
and c to a., what is the new expression? It is b+c+a à(2) Again,
suppose we change b to c, c to a and a to b, what is
the new expression? It is c+a+b à(3) We note
that the expressions (1), (2) and (3) are all same. 

Observe the expression ab+c à(1)
In the above expression,
suppose we change a to b, b to c and c to a., what is
the new expression?
It is bc+a
à(2)
Again, suppose we change b
to c, c to a and a to b, what is the new expression?
It is ca+b
à(3)
We note that the
expressions (1), (2) and (3) are not same unless
a=b=c
An expression f(x,y,z) is said to be cyclic with
respect to x,y,z if f(x,y,z)
= f(y,z,x) = f(z,x,y)
You will again notice that the expressions x^{2}+y^{2}+
z^{2} and x^{3}+y^{3} +z^{3 }are cyclic with
respect to variables x, y, z
The expression a+b+c is symbolically represented by
Similarly the expression x^{3}+y^{3}
+z^{3 }is symbolically represented by
2.16 Example 1:
Write a^{2}+b^{2}+ c^{2}abbcca using _{} notation
a^{2}+b^{2}+
c^{2}abbcca
= (a^{2}+b^{2}+ c^{2})(ab+bc+ca) = ( )or =()
2.16 Example 2:Expand
= xy(x^{2}y^{2} )+yz(y^{2}z^{2})+zx(z^{2}x^{2})
2.16
Summary of learning
No 
Points studied 
1 
Understanding
of cyclic symmetry 
Additional Points:
Factorisation of Cyclic expressions:
2.16 Problem 1:
Factorise ab(ab)+bc(bc)+ca(ca)
We use the following
identities for factorisation
No 
Formula/Identity 
Expansion 
1 
(a+b)^{2} 
a^{2}+b^{2}+2ab 
2 
(ab)^{2} 
a^{2}+b^{2}2ab 
3 
(a+b)(ab) 
a^{2}b^{2} 
4 
(a+b)^{3} 
a^{3}+b^{3}+3ab(a+b) 
5 
(ab)^{3} 
a^{3}b^{3}3ab(ab) 
6 
a^{3}+b^{3} 
(a+b) (a^{2} +b^{2} ab) 
7 
a^{3}b^{3} 
(ab) (a^{2} +b^{2} +ab) 
Solution:
ab(ab)+bc(bc)+ca(ca) = a^{2}b
 ab^{2}+bc(bc)+ c^{2}a  ca^{2 }(Expand
first and last two terms)
= a^{2}(bc)
 a(b^{2} c^{2}) + bc(bc) (Group those terms containing a^{2} and a
together)
= a^{2}(bc)  a(b+c)(bc) + bc(bc) (Factorise (b^{2}
c^{2}))
= (bc)(a^{2}a(b+c)+bc) (Taking out (bc) as common factor)
= (bc)(a^{2}abac+bc) (Group those terms containing a and c
together)
= (bc)(a(ab)c(ab))
= (bc)((ab)(ac)
=  (ab)(bc)(ca)
If all the terms of an
algebraic expression are of the same degree then such an expression is called a
Homogenous expression.
Ex. a^{2}+b^{2}+2ab
is a homogenous expression.
a^{2}+b^{2}+a is not a homogenous expression.