Toán Lớp 8: Chứng minh : Nếu $a^{3}$+ $b^{3}$+ $c^{3}$ =3abc thì a+b+c=0 hoặc a=b=c
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Toán Lớp 8: Chứng minh : Nếu $a^{3}$+ $b^{3}$+ $c^{3}$ =3abc thì a+b+c=0 hoặc a=b=c
Comments ( 2 )
⇔ a^3+3a^2b+3ab^2+b^3−(3a^2b+3ab^2)+c^3−3abc=0
⇔ (a+b)^3+c^3−3ab(a+b+c)=0
⇔ (a+b+c)[(a+b)^2−(a+b).c+c^2]−3ab(a+b+c)=0
⇔ (a+b+c)(a^2+b^2+c^2+2ab−ac−bc)−3ab(a+b+c)=0
⇔ (a+b+c)(a^2+b^2+c^2+2ab−ac−bc−3ab)=0
⇔ (a+b+c)(a^2+b^2+c^2−ac−bc−ab)=0
⇔ (a+b+c)(2a^2+2b^2+2c^2−2ac−2bc−2ab)=0
⇔ (a+b+c)[(a−b)^2+(b−c)^2+(c−b)^2]=0
⇔ \(\left[ \begin{array}{l}a+b+c=0\\(a−b)^2+(b−c)^2+(c−b)^2=0\end{array} \right.\)
⇔ \(\left[ \begin{array}{l}a+b+c=0\\a=b=c\end{array} \right.\) (đpcm)