Question

Alguém sabe essa ? Preciso com urgência !!

Answer 1

Sejam:

$u=b-c\\\\ v=c-a\\\\ w=a-b$

Substituindo na expressão E dada:

$E=\dfrac{2}{b\!-\!c}\!+\!\dfrac{b-c}{(c\!-\!a)\!(a\!-\!b)}\!+\!\dfrac{2}{c\!-\!a}\!+\!\dfrac{c-a}{(a\!-\!b)\!(b\!-\!c)}\!+\!\dfrac{2}{a\!-\!b}\!+\!\dfrac{a-b}{(b\!-\!c)\!(c\!-\!a)}\\\\ E=\dfrac{2}{u}+\dfrac{u}{vw}+\dfrac{2}{v}+\dfrac{v}{wu}+\dfrac{2}{w}+\dfrac{w}{uv}\\\\ E=\dfrac{2}{u}+\dfrac{2}{v}+\dfrac{2}{w}+\dfrac{u}{vw}+\dfrac{v}{wu}+\dfrac{w}{uv}\\\\ E=2\left(\dfrac{1}{u}+\dfrac{1}{v}+\dfrac{1}{w}\right)+\dfrac{u^2}{uvw}+\dfrac{v^2}{uvw}+\dfrac{w^2}{uvw}$

$E=2\left(\dfrac{vw}{uvw}+\dfrac{uw}{uvw}+\dfrac{uv}{uvw}\right)+\dfrac{u^2+v^2+w^2}{uvw}\\\\ E=\dfrac{u^2+v^2+w^2+2(uv+uw+vw)}{uvw}\\\\ E=\dfrac{(u+v+w)^2}{uvw}$

Note que a soma u+v+w do numerador equivale a:

$u+v+w=(c-a)+(b-c)+(a-c)=0$

Logo:

$E=\dfrac{(u+v+w)^2}{uvw}=\dfrac{0^2}{uvw}\\\\ \boxed{E=0}\Longrightarrow\text{Letra }\bold{E}.$