Question

1/(a*(a-b)*(a-c))+1/(b*(b-a)*(b-c))+1/(c*(c-b)*(c-a))

giúp mình với\

 

Answer 1

Ta có: \(A=\frac{1}{a\left(a-b\right)\left(a-c\right)}+\frac{1}{b\left(b-a\right)\left(b-c\right)}+\frac{1}{c\left(c-b\right)\left(c-a\right)}\)

\(\Rightarrow A=\frac{bc\left(b-c\right)+ac\left(c-a\right)+ab\left(a-b\right)}{abc\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)

Xét tử thức:

M= bc(b-c) + ac(c-a) + ab(a-b) = bc(b-c) - ac[(b-c)+(a-b)] + ab(a-b)

= bc(b-c) - ac(b-c) - ac(a-b) + ab(a-b) = c(b-c)(b-a) + a(a-b)(b-c)

= (a-b)(b-c)(a-c)

\(\Rightarrow A=\frac{\left(a-b\right)\left(b-c\right)\left(a-c\right)}{\left(a-b\right)\left(b-c\right)\left(a-c\right)abc}=\frac{1}{abc}\)