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

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

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

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

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

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

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

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

\(=VP\)

\(\Rightarrow\text{Điều phải chứng minh}\)