ta có \(a^3+b^3+a^2c+b^2c-abc=\left(a+b\right)\left(a^2-ab+b^2\right)+c\left(a^2-ab+b^2\right)=\left(a^2-ab+b^2\right)\left(a+b+c\right)\)

mà a+b+c=0

\(\Rightarrow a^3+b^3+a^2c+b^2c-abc=\left(a^2-ab+b^2\right).0=0\left(đpcm\right)\)

Ta có: a+b+c =0 => c= -a -b

Ta có a³ +a²c -abc + b²c +b³

= (a³ +b³) +c(a² -ab +b²)

= (a³ +b³) +(-a -b)(a² -ab +b²)

= (a³ +b³) -(a +b)(a² -ab +b²)

= (a³ +b³) -a³ -b³ = 0

Vậy a³ +a²c -abc +b²c +b³ =0

https://olm.vn/hoi-dap/detail/19699450579.html

Xem ở link này(mik gửi cho)

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a^3 + a^2c - abc + b^2c + b^3
= a^3+a^2c+a^2b-a^2b-abc+b^2c+b^3+b^2a-b^2a
= a^2(a+b+c)-a^2b-abc+b^2(a+b+c)-b^2a
= -a^2b-abc-b^2a
= -ab(a+b+c)=-ab*0 = 0
vậy đa thức này bằng 0

a+b+c=0
a^3 + a^2c - abc + b^2c + b^3 
=(a^3+a^2b+a^2c)-(a^2b+ab^2+abc)+(b^2c+b^3+ab^2)
=a^2(a+b+c)-ab(a+b+c)+b^2(a+b+c)
=0+0+0
=0

Có : \(a^3+a^2c-abc+b^2c+b^3\)

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

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

= ( a+b+c) ( \(a^2-ab+b^2\)) mà a+b+c=0

=> \(a^3+a^2c-abc+b^2c+b^3=0\left(đpcm\right)\)

\(a^3+a^2c-abc+b^2c+b^3=0\)

\(\Rightarrow a^2\left(a+c\right)-abc+b^2\left(b+c\right)=0\)

\(\Rightarrow-a^2b-abc-b^2a=0\)

\(\Rightarrow a^2b+abc+b^2a=0\)

\(\Rightarrow ab\left(a+b+c\right)=0\)(đúng)

a:

\(a^3+a^2c-abc+b^2c+b^3\)

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

\(=\left(a^2-ab+b^2\right)\left(a+b+c\right)=0\)(vì a+b=c=0)

câu b bn xem ở link này nha!

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\(a^3+a^2c-abc+b^2c+b^3\)

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

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

\(\Rightarrow\left(a+b+c\right)\left(a^2-ab+b^2\right)=0\)( vì a+b+c=0)

Vậy \(a^3+a^2c-abc+b^2c+b^3=0\left(đpcm\right)\)

\(b,A=bc\left(a+d\right)\left(b-c\right)-ac\left(b+d\right)\left(a-c\right)+ab\left(c+d\right)\left(a-b\right)\)

\(=bc\left(a+d\right)\left[\left(b-a\right)+\left(a-c\right)\right]-ac\left(a-c\right)\left(b+d\right)+ab\left(c+d\right)\left(a-b\right)\)

\(=-bc\left(a+d\right)\left(a-b\right)+bc\left(a+d\right)\left(a-c\right)-ac\left(a-c\right)\left(b+d\right)+ab\left(c+d\right)\left(a-b\right)\)

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

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

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

\(A=a^3+a^2c-abc+b^2c+b^3\\ =\left(a^3+b^3\right)+\left(a^2c-abc+b^2c\right)\\ =\left(a+b\right)\left(a^2-ab+b^2\right)+\left(a^2-ab+b^2\right)c\\ =\left(a+b+c\right)\left(a^2-ab+b^2\right)\\ Thay\text{ }a+b+c=0,\text{ }ta\text{ }được:\text{ }\\ A=\left(a+b+c\right)\left(a^2-ab+b^2\right)\\ =0\cdot\left(a^2-ab+b^2\right)\\ =0\)

Vậy \(A=0\) tại \(a+b+c=0\)

\(a^3+a^2c-abc+b^2c+b^3.\)

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

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

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

theo đề ta có \(a+b+c=0\)

\(\Rightarrow\left(a^2-ab+b^2\right)\left(a+b+c\right)\)

\(=\left(a^2-ab+b^2\right)\cdot0=0\)

\(\Rightarrow a^3+a^2c-abc+b^2c+b^3=0\left(đpcm\right)\)

Bài làm

Ta có: \(a^3+a^2c-abc+b^2c+b^3\)

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

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

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

Thay \(a+b+c=0\)và biểu thức trên ta được:

\(=0.\left(a^2-ab+b^2\right)\)

\(=0\)( đpcm )

~ Bài này khó v~, mất nửa tiếng ms nghĩ ra. ~
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