Question

Chứng minh đẳng thức:

\(\frac{b+c}{\left(a-b\right)\left(a-c\right)}+\frac{c+a}{\left(b-c\right)\left(b-a\right)}+\frac{a+b}{\left(c-a\right)\left(c-b\right)}=0\)

Answer 1

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

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

\(=\frac{b^2-c^2}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\frac{c^2-a^2}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\frac{a^2-b^2}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}\)

\(=\frac{b^2-c^2+c^2-a^2+a^2-b^2}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=0\left(ĐPCM\right)\)