Question

 Cho a,b,c khác nhau thõa mãn  \(a^2\left(b+c\right)=b^2\left(c+a\right)\) . Chứng minh : \(b^2\left(c+a\right)=c^2\left(a+b\right)\)

 

Answer 1

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

\(\Rightarrow a^2b+a^2c-b^2c-b^2a=0\)

\(\Rightarrow ab.\left(a-b\right)+c.\left(a-b\right).\left(a+b\right)=0\)

\(\Rightarrow\left(ab+ac+bc\right)\left(a-b\right)=0\)

Vậy : \(\left(ab+bc+ca\right)=0\)

\(\Rightarrow\left(ab+bc+ca\right).\left(b-c\right)=0\)

\(\Rightarrow b^2a+b^2c-c^2b-c^2a=0\)

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