Question

Chứng minh :

a, (a+b+c)³-a³-b³-c³=3(a+b)(b+c)(c+a)

Answer 1

`= a^3 + b^3 + c^3 + 3(a+b)(b+c)(c+a) - a^3 - b^3 - c^3 = 3(a+b)(b+c)(c+a)`

`=> 3(a+b)(b+c)(c+a) = 3(a+b)(b+c)(c+a) => dpcm`.

Answer 2

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

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

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

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

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