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A = ( a + b + c )3 + ( a - b - c )3 + ( b - c - a )3 + ( c - a - b )3
= [ ( a + b ) + c ]3 + [ ( a - b ) - c ]3 + [ ( - c ) - ( a - b ) ] 3 + [ c - ( a + b ) ]3
= ( a + b )3 + 3.( a + b )2.c + 3.( a + b ).c2 + c3 + ( a - b )3 - 3.( a - b )2.c + 3.( a - b ).c2 - c3 + ( - c3 ) + 3.( a - b )2.c - 3.( a - b ).c2 -(a- b)3
+ c3 + 3.( a + b )2.c - 3.( a + b ).c2 - ( a + b )3
= 6.( a + b )2 .c
(a+b+c)3−a3−b3−c3(a+b+c)3−a3−b3−c3
=a3+3a2(b+c)+3a(b+c)2+(b+c)3−a3−b3−c3=a3+3a2(b+c)+3a(b+c)2+(b+c)3−a3−b3−c3
=3(b+c)(a2+ab+ac)+b3+3b2c+3bc2+c3−b3−c3=3(b+c)(a2+ab+ac)+b3+3b2c+3bc2+c3−b3−c3
=3(b+c)(a2+ab+ac+bc)=3(b+c)(a2+ab+ac+bc)
=3(b+c)[a(a+b)+c(a+b)]=3(b+c)[a(a+b)+c(a+b)]
=3(b+c)(a+b)(a+c)
a) \(a^3+b^3+c^3-3abc\)
\(=\left(a^3+3a^2b+3ab^2+b^3\right)+c^3-3a^2b-3ab^2-3abc\)
\(=\left[\left(a+b\right)^3+c^3\right]-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left(a^2-ab+b^2-ac-bc+c^2\right)\)
b) \(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\)
\(=\left(x-y+y-z\right)\left(x^2-2xy+y^2-xy+xz+y^2-yz+y^2-2yz+z^2\right)+\left(z-x\right)^3\)
\(=\left(x-z\right)\left(x^2-3xy+2y^2+xz-3yz+z^2\right)-\left(x-z\right)^3\)
\(=\left(x-z\right)\left(x^2-3xy+2y^2+xz-3yz+z^2-x^2+2xz-z^2\right)\)
\(=\left(x-z\right)\left(-3xy+2y^2+3xz-3yz\right)\)
a) \(A=a^3-b^3-c^3-3abc\)
\(=\left(a-b\right)^3+3ab\left(a-b\right)-c^3-3abc\)
\(=\left(a-b-c\right)\left[\left(a-b\right)^2+c\left(a-b\right)+c^2\right]+3ab\left(a-b-c\right)\)
\(=\left(a-b-c\right)\left(a^2-2ab+b^2+ac-bc+c^2+3ab\right)\)
\(=\left(a-b-c\right)\left(a^2+b^2+c^2+ab+ac-bc\right)\)
b) \(B=a^2b^2\left(a-b\right)-c^2b^2\left(c-b\right)+a^2c^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+c^2b^2\left(b-c\right)+a^2c^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+c^2b^2\left(b-c\right)-a^2c^2\left[\left(a-b\right)+\left(b-c\right)\right]\)
\(=a^2b^2\left(a-b\right)+c^2b^2\left(b-c\right)-a^2c^2\left(a-b\right)-a^2c^2\left(b-c\right)\)
\(=a^2\left(a-b\right)\left(b^2-c^2\right)+c^2\left(b-c\right)\left(b^2-a^2\right)\)
\(=a^2\left(a-b\right)\left(b-c\right)\left(b+c\right)+c^2\left(b-c\right)\left(b-a\right)\left(b+a\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2b+a^2c-bc^2-ac^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\left(ab+bc+ca\right)\)
Đặt A là tên biểu thức; \(a+b-c=x;b+c-a=y;c+a-b=z\)
Khi đó \(x+y+z=a+b-c+b+c-a+c+a-b=a+b+c\)
=>\(A=\left(x+y+z\right)^3-x^3-y^3-z^3=\left[\left(x+y\right)+z\right]^3-x^3-y^3-z^3\)
\(=\left(x+y\right)^3+z^3+3z\left(x+y\right)\left(x+y+z\right)-x^3-y^3-z^3\)
\(=x^3+y^3+3xy\left(x+y\right)+z^3+3z\left(x+y\right)\left(x+y+z\right)-x^3-y^3-z^3\)
\(=3\left(x+y\right)\left(xy+xz+yz+z^2\right)\)
\(=3\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]=3\left(x+y\right)\left(y+z\right)\left(x+z\right)\)
\(=3\left(a+b-c+b+c-a\right)\left(b+c-a+c+a-b\right)\left(c+a-b+a+b-c\right)\)
\(=3.2b.2c.2a=24abc\)
a^3+b^3+c^3-3abc
=a^3+b^3+c^3-3abc+3a^2b-3a^2b+3ab^2-3ab^2
=(a+b)^3+c^3-3abc(a+b+c)
=(a+b+c)[(a+b)^2-(a+b)c+c^2]-3ab(a+b+c)
=(a+b+c)(a^2+2ab+b^2-ac-bc+c^2-3ab)
=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)
nhớ tích cho mạnh nhé !!