Question

encontre o valor da expressao a2/(a-b)(a-c)+b2/(b-a)(b-c)+c2/(c-a)(c-b)??

Answer 1

Essa primeira parte será utilizada lá no final.
   (a - b) · (a - c) · (b - c)
= (a² - ac - ab + bc) · (b - c)
= a²b - a²c - abc + ac² - ab² + abc + b²c - bc²
= a²b - a²c + ac² - ab² - abc + abc + b²c - bc²
= a²b - a²c + ac² - ab² + b²c - bc²
= a²b - a²c - ab² + b²c + ac² - bc²
= a² · (b - c) - b² · (a - c) + c² · (a - b)

Concluímos que:
(a - b) · (a - c) · (b - c) = a² · (b - c) - b² · (a - c) + c² · (a - b)
---------------------------------------------------------------------------------------------------------
$\ \ \ \frac{a^2}{(a-b)\cdot(a-c)}+\frac{b^2}{(b-a)\cdot (b-c)}+\frac{c^2}{(c-a)\cdot(c-b)}\\ \\=\frac{a^2}{(a-b)\cdot(a-c)}+\frac{b^2}{-(-b+a)\cdot (b-c)}+\frac{c^2}{-(-c+a)\cdot-(-c+b)}\\ \\=\frac{a^2}{(a-b)\cdot(a-c)}+\frac{b^2}{-(a-b)\cdot (b-c)}+\frac{c^2}{-(a-c)\cdot-(b-c)}\\ \\=\frac{a^2}{(a-b)\cdot(a-c)}-\frac{b^2}{(a-b)\cdot (b-c)}+\frac{c^2}{(a-c)\cdot(b-c)}$
$\\=\frac{a^2\cdot(b-c)}{(a-b)\cdot(a-c)\cdot(b-c)}-\frac{b^2\cdot(a-c)}{(a-b)\cdot(a-c)\cdot(b-c)}+\frac{c^2\cdot(a-b)}{(a-b)\cdot(a-c)\cdot(b-c)}\\ \\=\frac{a^2\cdot(b-c)-b^2\cdot(a-c)+c^2\cdot(a-b)}{(a-b)\cdot(a-c)\cdot(b-c)} =\frac{(a-b)\cdot(a-c)\cdot(b-c)}{(a-b)\cdot(a-c)\cdot(b-c)}=1$