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a+b+c+d=0
=> a + b = -(c+d)
=> (a+b)^3 = -(c+d)^3
=> a^3 + b^3 + 3ab (a+b) = -c^3- d^3 - 3cd (c+d)
=> a^3+b^3+c^3+d^3 = -3ab (a+b) - 3cd (c+d)
=> a^3 + b^3 + c^3 + d^3 = 3ab (c+d)- 3cd (c+d) [vì a+b = - (c+d)]
==> a^3 + b^^3 + c^3 + d^3 =3 (c+d) (ab-cd) (đpcm)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
ta có a+b+c+d = 0=> b+c= -( a+d) => (b+c)^3 = - (a+d)^3
=> b^3+ c^3 + 3bc( b+c) = -( a^3 +d^3 + 3ad(a+d))
=> a^3+b^3+c^3+d^3 = - 3ad( a+d) - 3bc(b+c) = 3ad(b+c) - 3bc(b+c)
= 3(b+c)(ad-bc)
sao cậu tự đặt câu hỏi rồi lại tự trả lời luôn
thế là sao??????????
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có :
\(a+b+c+d=0\)
\(\Rightarrow b+c=-\left(a+d\right)\)
\(\Rightarrow\left(b+c\right)^2=\left(a+d\right)^2\)
\(\Rightarrow\left(b+c\right)^2-\left(a+d\right)^2=0\)
\(\Rightarrow b^2+c^2+2bc-a^2-d^2-2ad=0\)
Lại có :
\(a^3+b^3+c^3+d^3\)
\(=\left(a+d\right)\left(a^2+d^2-ad\right)+\left(b+c\right)\left(b^2+c^2-bc\right)\)
\(=\left(b+c\right)\left(b^2+c^2-bc\right)-\left(b+c\right)\left(a^2+d^2-ad\right)\)
\(=\left(b+c\right)\left[\left(b^2+c^2-bc\right)-\left(a^2+d^2-ad\right)\right]\)
\(=\left(b+c\right)\left[\left(b^2+c^2+2bc-a^2-d^2-2ad\right)+3ad-3bc\right]\)
\(=\left(b+c\right)\left[0+3\left(ad-bc\right)\right]\)
\(=3\left(b+c\right)\left(ad-bc\right)\)
Vậy ...
Ta có : a + b +c + d = 0
=> a + d = - b - c
=> (a + d) = -(b + c)
=> (a + d)3 = -(b + c)3
a3 + 3a2d + 3ad2 + d3 = -(b3 + 3b2c + 3bc2 + c3)
a3 + 3a2d + 3ad2 + d3 = -b3 - 3b2c - 3bc2 - c3
a3 + b3 + c3 + d3 = -3a2d - 3ad2 - 3b2c - 3bc2
a3 + b3 + c3 + d3 = -3ad(a + d) - 3bc(b + c)
a3 + b3 + c3 + d3 = -3ad(-b - c) - 3bc(b + c)
a3 + b3 + c3 + d3 = 3ad(b + c) - 3bc(b + c)
a3 + b3 + c3 + d3 = 3(b + c)(ad - bc)