Question

Chứng minh:

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

Answer 1

Ta có: \(VT=\left(a+b+c\right)^3-a^3-b^3-c^3\)

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

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

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

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

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

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

P/s: Bài nhà cô Yến à m? :">