K
Khách

Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.

a+b+c+d=0

nên a+b=-(c+d)

\(a^3+b^3+c^3+d^3\)

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

\(=\left[-\left(c+d\right)\right]^3-3ab\cdot\left[-\left(c+d\right)\right]+\left(c+d\right)^3-3cd\left(c+d\right)\)

\(=3ab\left(c+d\right)-3cd\left(c+d\right)\)

\(=3\left(c+d\right)\left(ab-cd\right)\)

15 tháng 9 2018

2 ) b )

\(a+b+c+d=0\)

\(\Leftrightarrow a+b=-\left(c+d\right)\)

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

\(\Leftrightarrow a^3+b^3+3a^2b+3b^2a=-c^3-3c^2d-3d^2c-d^3\)

\(\Leftrightarrow a^3+b^3+3a^2b+3b^2a+c^3+3c^2d+3d^2c+d^3=0\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=-3a^2b-3b^2a-3c^2d-3d^2c\)

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

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

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

5 tháng 7 2023

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

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

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

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

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

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

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

2 tháng 7 2017

Giải:

Từ \(a+b+c+d=0\Leftrightarrow a+c=-\left(b+d\right)\)

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

\(\Leftrightarrow a^3+c^3+3ac\left(a+c\right)=-\left[b^3+d^3+3bd\left(b+d\right)\right]\)

\(\Leftrightarrow VT=a^3+b^3+c^3+d^3=-3bd\left(b+d\right)-3ac\left(a+c\right)\)

\(=-3bd\left(b+d\right)+3ac\left(b+d\right)=3\left(ac-bd\right)\left(b+d\right)=VP\) (Đpcm)

31 tháng 3 2020

Theo đề, a+b+c+d=0

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

Ta có: \(VT=\left(a+b\right)\left(a^2-ab+b^2\right)+\left(c+d\right)\left(c^2-cd+d^2\right)\)

\(\Leftrightarrow VT=\left(c+d)\left(c^2-cd+d^2-a^2+ab-b^2\right)\right)\)

Để có ĐPCM ta xét hiệu: \(c^2-cd+d^2-a^2+ab-b^2-3\left(ab+cd\right)=c^2-4cd+d^2-a^2-2ab-b^2=c^2-4cd+d^2-\left(a+b\right)^2=c^2-4cd+d^2-\left(c+d\right)^2=-6cd\)

S nó ko = 0 ta:::xem lại đề..Hay mk lm sai j đó

18 tháng 9 2018

a ) Ta có : \(a+b+c=0\)

\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+ac+bc\right)=0\)

\(\Leftrightarrow a^2+b^2+c^2=-2\left(ab+ac+bc\right)\)

\(\Leftrightarrow\left(a^2+b^2+c^2\right)^2=4\left(ab+ac+bc\right)^2\)

\(\Leftrightarrow a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2=4\left(a^2b^2+b^2c^2+c^2a^2+2ab^2c+2a^2bc+2c^2ab\right)\)

\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)+8abc\left(a+b+c\right)\)

\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+a^2c^2\right)+8abc.0\)

\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+a^2c^2\right)\)

Lại có : \(\dfrac{\left(a^2+b^2+c^2\right)^2}{2}=\dfrac{a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)}{2}\)

\(=\dfrac{a^4+b^4+c^4+a^4+b^4+c^4}{2}=\dfrac{2\left(a^4+b^4+c^4\right)}{2}\)

\(=a^4+b^4+c^4\left(đpcm\right)\)

18 tháng 9 2018

b ) \(a+b+c+d=0\)

\(\Leftrightarrow a+b=-\left(c+d\right)\)

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

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

\(\Leftrightarrow a^3+b^3+c^3+d^3+3a^2b+3b^2a+3c^2d+3d^2c=0\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=-3a^2b-3b^2a-3c^2d-3d^2c\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left(-a^2b-b^2a-c^2d-d^2c\right)\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left[-ab\left(a+b\right)-cd\left(c+d\right)\right]\)

\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left[ab\left(c+d\right)-cd\left(c+d\right)\right]\)

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

15 tháng 7 2019

Sai đề rồi bạn nhé

16 tháng 7 2019

Vậy đề đúng là gì bạn

29 tháng 6 2018

\(a.a^3+b^3+c^3=3abc\)

\(\Leftrightarrow\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0\)

Nếu : \(a+b+c=0\) thì đẳng thức trên đúng .

\(\Rightarrowđpcm\)

\(b.a+b+c+d=0\Rightarrow a+b=-\left(c+d\right)\Leftrightarrow\left(a+b\right)^3=-\left(c+d\right)^3\)

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

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

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

29 tháng 9 2019

a ) \(a^3+b^3+c^3=3abc\)

\(\Leftrightarrow\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0\)

Nếu : \(a+b+c=0\) thì đẳng thức trên đúng .

\(\Rightarrowđpcm\)

b ) \(a+b+c+d=0\)

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

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

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

\(\Leftrightarrow a^3+b^3+c^3+d^3=3\left(c+d\right)\left(cb-cd\right)\left(đpcm\right)\)

Chúc bạn học tốt !!!

29 tháng 9 2019

a ) a^3+b^3+c^3=3abca3+b3+c3=3abc

\Leftrightarrow\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0⇔(a+b)3+c3−3ab(a+b)−3abc=0

\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0⇔(a+b+c)(a2+b2+c2−ab−bc−ac)=0

Nếu : a+b+c=0a+b+c=0 thì đẳng thức trên đúng .(đpcm)

b ) a+b+c+d=0a+b+c+d=0

\Rightarrow a+b=-\left(c+d\right)\Leftrightarrow\left(a+b\right)^3=-\left(c+d\right)^3⇒a+b=−(c+d)⇔(a+b)3=−(c+d)3

\Leftrightarrow a^3+b^3+c^3+d^3=-3ab\left(a+b\right)-3cd\left(c+d\right)⇔a3+b3+c3+d3=−3ab(a+b)−3cd(c+d)

\Leftrightarrow a^3+b^3+c^3+d^3=3ab\left(c+d\right)-3cd\left(c+d\right)⇔a3+b3+c3+d3=3ab(c+d)−3cd(c+d)

\Leftrightarrow a^3+b^3+c^3+d^3=3\left(c+d\right)\left(cb-cd\right)\left(đpcm\right)⇔a3+b3+c3+d3=3(c+d)(cb−cd)(đpcm)