A=\(\dfrac{1}{\left(a-b\right)\left(a-c\right)}\)\(-\dfrac{1}{\left(a-b\right)\left(b-c\right)}\)\(+\dfrac{1}{\left(a-c\right)\left(b-c\right)}\)
<=>A=\(\dfrac{b-c-\left(a-c\right)+a-b}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)
<=> A=\(\dfrac{0}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)<=> A=0
Đặt: \(a-b=x\)
\(a-c=y\)
\(b-c=z\)
Ta có: \(A=\dfrac{1}{\left(a-b\right)\left(a-c\right)}+\dfrac{1}{\left(b-a\right)\left(b-c\right)}+\dfrac{1}{\left(c-a\right)\left(c-b\right)}\)
\(=\dfrac{1}{xy}-\dfrac{1}{xz}+\dfrac{1}{yz}\)
\(=\dfrac{xyz^2-xy^2z+x^2yz}{x^2y^2z^2}\)
\(=\dfrac{xyz\left(z-y+x\right)}{x^2y^2z^2}\)
\(=\dfrac{z-y+x}{xyz}\)
Thay \(a-b=x;a-c=y;b-c=z\) vào biểu thức \(\dfrac{z-y+x}{xyz}\), ta được:
\(\dfrac{\left(b-c\right)-\left(a-c\right)+\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
= \(\dfrac{b-c-a+c+a-b}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)
= 0
Vậy:\(A=\dfrac{1}{\left(a-b\right)\left(a-c\right)}+\dfrac{1}{\left(b-a\right)\left(b-c\right)}+\dfrac{1}{\left(c-a\right)\left(c-b\right)}=0\)
