Chứng minh hằng đẳng thức: (a+b+c)^3= a^3 + b^3 + c^3 + 3(a+b)(b+c)(c+a)

* Hướng dẫn giải

Biến đổi vế trái:

${\left(a+b+c\right)}^3$= ${\left[\left(a+\right)+c\right]}^3$ = ${\left(a+b\right)}^3$+3${\left(a+b\right)}^2$ c+3(a+b)$c^2$+$c^3$

      = $a^3$ + 3$a^2$b + 3a$b^2$ + $b^3$ + 3($a^2$ + 2ab + $b^2$)c + 3a$c^2$ + 3b$c^2$ + $c^3$

      = $a^3$ + 3$a^2$b + 3a$b^2$ + $b^3$ + 3$a^2$c + 6abc + 3$b^2$c + 3a$c^2$ + 3b$c^2$ + c3

      = $a^3$ + $b^3$ + $c^3$ + 3$a^2$b + 3a$b^2$ + 3$a^2$c + 6abc + 3$b^2$c + 3a$c^2$ + 3b$c^2$

      = $a^3$ + $b^3$ + $c^3$ + (3$a^2$b + 3a$b^2$) +( 3$a^2$c + 3abc)+ (3abc + 3$b^2$c)+(3a$c^2$ + 3b$c^2$)

      = $a^3$ + $b^3$ + $c^3$ + 3ab(a + b) + 3ac(a + b) + 3bc(a + b) + 3$c^2$(a + b)

      = $a^3$ + $b^3$ + $c^3$ + 3(a + b)(ab + ac + bc + $c^2$)

      = $a^3$ + $b^3$ + $c^3$ + 3(a + b)[a(b + c) + c(b + c)]

      = $a^3$ + $b^3$ + $c^3$ + 3(a + b)(b + c)(a + c) (đpcm)