Question

Expansion of $a^3+b^3+c^3-3abc$ is:

• A. $\frac{1}{2}(a+b+c)\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$
• B. $(a+b+c)(a^2+b^2+c^2-ab-bc-ca$
• C. Both (A) & (B)
• D. None of the above

Answer 1

The correct option is C Both (A) & (B)

$a^3+b^3+c^3-3abc$
$=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

$=\frac{1}{2}(a+b+c)(2a^2+2b^2+2c^2-2ab-2bc-2ca)$

$=\frac{1}{2}(a+b+c)[a^2+b^2-2ab+b^2+c^2-2bc+c^2+a^2-2ca]$

$=\frac{1}{2}(a+b+c)\left[\right(a^2+b^2-2ab)+(b^2+c^2-2bc)+(c^2+a^2-2ca\left)\right]$

$=\frac{1}{2}(a+b+c)\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$