Question

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

Answer 1

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

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

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

\(=a^3+b^3+c^3+3a^2b+3b^2a+3a^2c+6abc+3b^2c+3ac^2+3bc^2\)

\(=a^3+b^3+c^3+3\left(a^2b+b^2a+a^2c+2abc+b^2c+ac^2+bc^2\right)\)

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

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

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

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

\(\left(đpcm\right)\)

Answer 2

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

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

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

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

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

...........................\(=a^3+b^3+c^3+3\left(a+b\right)\left[a\left(b+c\right)+c\left(b+c\right)\right]\)

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