Question

cmr

(a+b+c)³=a³+b³+c³+3(a+b)(b+c)(a+c)

Answer 1

\(VP=a^3+b^3+c^3+3\left(a+b\right)\left(a+c\right)\left(b+c\right)\)

\(VP=\left(a+b\right)\left(a^2-ab+b^2\right)+c^3+3\left(a +b\right)\left(a+c\right)\left(b+c\right)\)

\(=\left(a+b\right)\left(a^2+2ab+b^2-3ab\right)+c^3+3\left(a+b\right)\left(a+c\right)\left(b+c\right)\)

\(=\left(a+b\right)^3-3ab\left(a+b\right)+c^3+3\left(a+b\right)\left(a+c\right)\left(b+c\right)\)

\(=\left(a+b\right)^3+c^3-3\left(a+b\right)\left[\left(ab+ac+bc+c^2\right)\right]\)

\(=\left(a+b+c\right)\left[\left(a+b\right)^2-ac-bc+c^2\right]-3\left(a+b\right)\left(a+b\right)\left(a+c\right)\)