Câu hỏi của Thanh Tu Nguyen

Question

Cho : a + b + c + d = 0

Chứng minh rằng $a^3+b^3+c^3+d^3=3\left(ab-cd\right)\left(c+d\right)$

Đinh Trần Minh · Accepted Answer

$a+b+c+d=0\Rightarrow a+b=-\left(c+d\right)$

$\Rightarrow\left(a+b\right)^3=-\left(c+d\right)^3$

$\Rightarrow\left(a+b\right)^3+\left(c+d\right)^3=0$

$\Rightarrow a^3+b^3+3ab\left(a+b\right)+c^3+d^3+3cd\left(c+d\right)=0$

$\Rightarrow a^3+b^3+c^3+d^3=-3ab\left(a+b\right)-3cd\left(c+d\right)$

$\Rightarrow a^3+b^3+c^3+d^3=3ab\left(c+d\right)-3cd\left(c+d\right)$ (do $a+b=-\left(c+d\right)$

$\Rightarrow a^3+b^3+c^3+d^3=3\left(ab-cd\right)\left(c+d\right)$Câu trả lời: $a+b+c+d=0\Rightarrow a+b=-\left(c+d\right)$

$\Rightarrow\left(a+b\right)^3=-\left(c+d\right)^3$

$\Rightarrow\left(a+b\right)^3+\left(c+d\right)^3=0$

$\Rightarrow a^3+b^3+3ab\left(a+b\right)+c^3+d^3+3cd\left(c+d\right)=0$

$\Rightarrow a^3+b^3+c^3+d^3=-3ab\left(a+b\right)-3cd\left(c+d\right)$

$\Rightarrow a^3+b^3+c^3+d^3=3ab\left(c+d\right)-3cd\left(c+d\right)$ (do $a+b=-\left(c+d\right)$

$\Rightarrow a^3+b^3+c^3+d^3=3\left(ab-cd\right)\left(c+d\right)$