Question

Simplify the following cyclic expressions.

$\sum _{a,b,c}(b-c)(b+c)$

Answer 1

Cyclic expressions refers to substitution of variables in cyclic manner.

We have, ${\sum }_{a,b,c}(b-c)(b+c)$

Here, a,b,c are variables & the two expressions are multiplied.

So, ${\sum }_{a,b,c}(b-c)(b+c)=(b-c)(b+c)+(c-a)(c+a)+(a-b)(a+b)$

$=b^2-c^2+c^2-a^2+a^2-b^2[\because x^2-y^2=(x-y\left)\right(x+y\left)\right]$

$=0$

Hence, ${\sum }_{a,b,c}(b-c)(b+c)=0$