Biết a + b + c = 0. Chứng minh a^3 + b^3 + c^3 = 3abc. 05/09/2021 Bởi Iris Biết a + b + c = 0. Chứng minh a^3 + b^3 + c^3 = 3abc.
\(a+b+c=0\) \(\Leftrightarrow a+b=-c\) \(\Leftrightarrow\left(a+b\right)^3=\left(-c\right)^3\) \(\Leftrightarrow\left(a+b\right)^3+c^3=0\) \(\Leftrightarrow a^3+b^3+3ab\left(a+b\right)+c^3=0\) \(\Leftrightarrow a^3+b^3+c^3+3ab.\left(-c\right)=0\) \(\Leftrightarrow a^3+b^3+c^3-3abc=0\Leftrightarrow a^3+b^3+c^3=3abc\) Bình luận
Ta có : $a+b+c=0$ $\to a+b=-c$ $\to (a+b)^3=(-c)^3$ $\to a^3+b^3+c^3=-3ab.(a+b)$ $ \to a^3+b^3+c^3=3abc$ Bình luận
\(a+b+c=0\)
\(\Leftrightarrow a+b=-c\)
\(\Leftrightarrow\left(a+b\right)^3=\left(-c\right)^3\)
\(\Leftrightarrow\left(a+b\right)^3+c^3=0\)
\(\Leftrightarrow a^3+b^3+3ab\left(a+b\right)+c^3=0\)
\(\Leftrightarrow a^3+b^3+c^3+3ab.\left(-c\right)=0\)
\(\Leftrightarrow a^3+b^3+c^3-3abc=0\Leftrightarrow a^3+b^3+c^3=3abc\)
Ta có :
$a+b+c=0$
$\to a+b=-c$
$\to (a+b)^3=(-c)^3$
$\to a^3+b^3+c^3=-3ab.(a+b)$
$ \to a^3+b^3+c^3=3abc$