Question

State True or False.

$(a+b{)}^3+(b+c{)}^3+(c+a{)}^3-3(a+b\left)\right(b+c\left)\right(c+a)=2(a^3+b^3+c^3-3abc)$

• A. True
• B. False

Answer 1

The correct option is A True

$L.H.S:(a+b{)}^3+(b+c{)}^3+(c+a{)}^3-3(a+b\left)\right(b+c\left)\right(c+a)$
$=(a^3+3a^{2}b+3a{b}^2+b^3)+(b^3+3b^{2}c+3b{c}^2+c^3)+(c^3+3c^{2}a+3c{a}^2+a^3)-3(a+b)(bc+ba+c^2+ca)$
$=(2a^3+2b^3+2c^3+3a^{2}b+3a{b}^2+3b^{2}c+3b{c}^2+3c^{2}a+3c{a}^2)-3(a+b)(bc+ba+c^2+ca)$
$=(2a^3+2b^3+2c^3+3a^{2}b+3a{b}^2+3b^{2}c+3b{c}^2+3c^{2}a+3c{a}^2)-3(abc+b{a}^2+a{c}^2+c{a}^2+b^{2}c+b^{2}a+b{c}^2+abc)$
$=2a^3+2b^3+2c^3-6abc$
$=2(a^3+b^3+c^3-3abc)$
$=R.H.S$