Question

Resolve the following into factors:

$\underset{a,b,c}{(\sum a{)}^3}-\underset{a,b,c}{(\sum a^3)}$

• A. $3(a+b)(b+c)(c+a)$
• B. $-3(a+b)(b+c)(c+a)$
• C. $3(a-b)(b+c)(c-a)$
• D. None

Answer 1

The correct option is A $3(a+b)(b+c)(c+a)$

Cyclic expressions refers to substitution of variables in cyclic manner.

We have, ${\left({\sum }_{a,b,c}a\right)}^3-\left({\sum }_{a,b,c}{a}^3\right)$

$=(a+b+c{)}^3-(a^3+b^3+c^3)$

$=[a^3+b^3+c^3+3(a+b\left)\right(b+c\left)\right(c+a\left)\right]-(a^3+b^3+c^3)[\because (x+y+z{)}^3=x^3+y^3+z^3+3(x+y)(y+z)(z+x)]$

$=3(a+b)(b+c)(c+a)$

So, $\text{A}$ is the correct option.