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Lời giải:
a)
$yz(y+z)+xz(z-x)-xy(x+y)=yz(y+z)+xz^2-x^2z-x^2y-xy^2$
$=yz(y+z)+x(z^2-y^2)-x^2(z+y)$
$=yz(y+z)+x(z-y)(z+y)-x^2(z+y)$
$=(y+z)(yz+xz-xy-x^2)$
$=(y+z)[z(x+y)-x(x+y)]=(y+z)(x+y)(z-x)$
b)
$2a^2b+4ab^2-a^2c+ac^2-4b^2c+2bc^2-4abc$
$=(2a^2b+4ab^2)-(a^2c+2abc)+(ac^2+2bc^2)-(4b^2c+2abc)$
$=2ab(a+2b)-ac(a+2b)+c^2(a+2b)-2bc(a+2b)$
$=(a+2b)(2ab-ac+c^2-2bc)$
$=(a+2b)[2b(a-c)-c(a-c)]$
$=(a+2b)(2b-c)(a-c)$
c)
$y(x-2z)^2+8xyz+x(y-2z)^2-2z(x+y)^2$
$=y[(y-2z)+(x-y)]^2+8xyz+x(y-2z)^2-2z(x+y)^2$
$=y(y-2z)^2+y(x-y)^2+2y(y-2z)(x-y)+8xyz+x(y-2z)^2-2z(x+y)^2$
$=y(y-2z)^2+y(x+y)^2-4xy^2+2y(y-2z)(x-y)+8xyz+x(y-2z)^2-2z(x+y)^2$
$=(y-2z)^2(x+y)+(x+y)^2(y-2z)-4xy(y-2z)+2y(y-2z)(x-y)$
$=(y-2z)^2(x+y)+(x+y)^2(y-2z)+2y(y-2z)(x-y-2x)$
$=(y-2z)^2(x+y)+(x+y)^2(y-2z)-2y(y-2z)(x+y)$
$=(x+y)(y-2z)[(y-2z)+(x+y)-2y]=(x+y)(y-2z)(x-2z)$
Lời giải:
a)
$yz(y+z)+xz(z-x)-xy(x+y)=yz(y+z)+xz^2-x^2z-x^2y-xy^2$
$=yz(y+z)+x(z^2-y^2)-x^2(z+y)$
$=yz(y+z)+x(z-y)(z+y)-x^2(z+y)$
$=(y+z)(yz+xz-xy-x^2)$
$=(y+z)[z(x+y)-x(x+y)]=(y+z)(x+y)(z-x)$
b)
$2a^2b+4ab^2-a^2c+ac^2-4b^2c+2bc^2-4abc$
$=(2a^2b+4ab^2)-(a^2c+2abc)+(ac^2+2bc^2)-(4b^2c+2abc)$
$=2ab(a+2b)-ac(a+2b)+c^2(a+2b)-2bc(a+2b)$
$=(a+2b)(2ab-ac+c^2-2bc)$
$=(a+2b)[2b(a-c)-c(a-c)]$
$=(a+2b)(2b-c)(a-c)$
c)
$y(x-2z)^2+8xyz+x(y-2z)^2-2z(x+y)^2$
$=y[(y-2z)+(x-y)]^2+8xyz+x(y-2z)^2-2z(x+y)^2$
$=y(y-2z)^2+y(x-y)^2+2y(y-2z)(x-y)+8xyz+x(y-2z)^2-2z(x+y)^2$
$=y(y-2z)^2+y(x+y)^2-4xy^2+2y(y-2z)(x-y)+8xyz+x(y-2z)^2-2z(x+y)^2$
$=(y-2z)^2(x+y)+(x+y)^2(y-2z)-4xy(y-2z)+2y(y-2z)(x-y)$
$=(y-2z)^2(x+y)+(x+y)^2(y-2z)+2y(y-2z)(x-y-2x)$
$=(y-2z)^2(x+y)+(x+y)^2(y-2z)-2y(y-2z)(x+y)$
$=(x+y)(y-2z)[(y-2z)+(x+y)-2y]=(x+y)(y-2z)(x-2z)$
\(yz\left(y+z\right)+zx\left(z-x\right)-xy\left(x+y\right)\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left[\left(y+z\right)-\left(z-x\right)\right]\)
\(=yz\left(y+z\right)+zx\left(z-x\right)-xy\left(y+z\right)+xy\left(z-x\right)\)
\(=y\left(y+z\right)\left(z-x\right)+x\left(z-x\right)\left(z-y\right)\)
\(=\left(z-x\right)\left(yz-xy+xz-xy\right)\)
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) \(B=a^2b^2\left(a-b\right)-c^2b^2\left(c-b\right)+a^2c^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+c^2b^2\left(b-c\right)+a^2c^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+c^2b^2\left(b-c\right)-a^2c^2\left[\left(a-b\right)+\left(b-c\right)\right]\)
\(=a^2b^2\left(a-b\right)+c^2b^2\left(b-c\right)-a^2c^2\left(a-b\right)-a^2c^2\left(b-c\right)\)
\(=a^2\left(a-b\right)\left(b^2-c^2\right)+c^2\left(b-c\right)\left(b^2-a^2\right)\)
\(=a^2\left(a-b\right)\left(b-c\right)\left(b+c\right)+c^2\left(b-c\right)\left(b-a\right)\left(b+a\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a^2b+a^2c-bc^2-ac^2\right)\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\left(ab+bc+ca\right)\)
b) a3 + b3 + c3 - 3abc
= ( a + b)3 - 3ab - 3ba + c - 3abc
= (a3 + 3a2b + 3ab2 + b3) + c3 - (3a2b + 3ab2 + 3ab)
= (a + b)3 + c2 - 3ab(a + b + c)
= (a + b + c) [ (a + b)2 - ( a + b )c + c^2 ] - 3ab(a + b + c)
= ( a + b + c ) ( a2 + b2 + 2ab - ac - bc + c2 -3ab )
= ( a + b + c ) ( a2 + b2 + c2 - ab - ac - bc