Question

Phân tích đa thức thành nhân tử \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{a+b+c}\)

Answer 1

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{a+b+c}=\left(\frac{1}{a}+\frac{1}{b}\right)+\left(\frac{1}{c}-\frac{1}{a+b+c}\right)\)

\(=\frac{a+b}{ab}+\frac{a+b}{c\left(a+b+c\right)}=\left(a+b\right)\left(\frac{1}{ab}+\frac{1}{c\left(a+b+c\right)}\right)=\frac{\left(a+b\right)\left(ab+ac+bc+c^2\right)}{abc\left(a+b+c\right)}\)

\(=\frac{\left(a+b\right)\left(a+c\right)\left(b+c\right)}{abc\left(a+b+c\right)}\)