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 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 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 x2+y2+ z2 and x3+y3
+z3 are cyclic with respect to variables x, y, z
The expression a+b+c is
symbolically represented by 
Similarly the expression x3+y3 +z3
is symbolically represented by
2.16
Example 1: Write
a2+b2+ c2-ab-bc-ca using 
notation
a2+b2+ c2-ab-bc-ca = (a2+b2+
c2)-(ab+bc+ca) =
( 
)or =(
)
2.16
Example 2:Expand 
= xy(x2-y2 )+yz(y2-z2)+zx(z2-x2)
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(a-b)+bc(b-c)+ca(c-a)
We use the following identities for factorisation
|
No |
Formula/Identity |
Expansion |
|
1 |
(a+b)2 |
a2+b2+2ab |
|
2 |
(a-b)2 |
a2+b2-2ab |
|
3 |
(a+b)(a-b) |
a2-b2 |
|
4 |
(a+b)3 |
a3+b3+3ab(a+b) |
|
5 |
(a-b)3 |
a3-b3-3ab(a-b) |
|
6 |
a3+b3 |
(a+b) (a2 +b2 -ab) |
|
7 |
a3-b3 |
(a-b) (a2 +b2 +ab) |
Solution:
ab(a-b)+bc(b-c)+ca(c-a)
= a2b - ab2+bc(b-c)+
c2a - ca2
(Expand first and last two terms)
= a2(b-c) - a(b2- c2)
+ bc(b-c) (Group those terms containing a2
and a together)
= a2(b-c)
- a(b+c)(b-c) + bc(b-c) (Factorise (b2- c2))
= (b-c)(a2-a(b+c)+bc)
(Taking out (b-c) as common factor)
= (b-c)(a2-ab-ac+bc)
(Group those terms containing a and c together)
= (b-c)(a(a-b)-c(a-b))
= (b-c)((a-b)(a-c)
= - (a-b)(b-c)(c-a)
If all the terms of an algebraic expression are of the
same degree then such an expression is called a Homogenous
expression.
Ex. a2+b2+2ab is
a homogenous expression.
a2+b2+a is not a homogenous expression.