Question

chứng minh: (a + b + c)(ab + bc + ca) = abc + (a + b)(b + c)(c + a)

Answer 1

Ta có \(\left(a+b+c\right)\left(ab+bc+ca\right)=a^2b+abc+a^2c+ab^2+b^2c+abc+abc+bc^2+ac^2=a^2b+ab^2+a^2c+ac^2+b^2c+bc^2+3abc\left(1\right)\)

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

Từ (1),(2)\(\Rightarrow\left(a+b+c\right)\left(ab+bc+ca\right)=abc+\left(a+b\right)\left(b+c\right)\left(c+a\right)\)