a: =(x+y)^3+z^3-3xy(x+y)-3xyz

=(x+y+z)(x^2+2xy+y^2-xz-yz+z^2)-3xy(x+y+z)

=(x+y+z)(x^2+y^2+z^2-xy-xz-yz)

b: a+b+c<>0

A=(a+b+c)^3-a^3-b^3-c^3/a+b+c

=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)/(a+b+c)

=a^2+b^2+c^2-ab-ac-bc

=1/2[a^2-2ab+b^2+b^2-2bc+c^2+a^2-2ac+c^2]

=1/2[(a-b)^2+(b-c)^2+(a-c)^2]>=0

a: =(x+y)^3+z^3-3xy(x+y)-3xyz

\(=\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)\)

b: \(=\left(x+y+y-z\right)^3-3\left(x+y\right)\left(y-z\right)\left(x+y+y-z\right)+\left(z-x\right)^3\)

\(=\left(x-z\right)^3+\left(z-x\right)^3-3\left(x+y\right)\left(y-z\right)\left(x-z\right)\)

\(=-3\left(x+y\right)\left(y-z\right)\left(x-z\right)\)

c: \(=\left(x^2+x\right)^2+3\left(x^2+x\right)+2-12\)

\(=\left(x^2+x\right)^2+3\left(x^2+x\right)-10\)

=(x^2+x+5)(x^2+x-2)

=(x^2+x+5)(x+2)(x-1)

d: =b^2c+bc^2+ac^2-a^2c-a^2b-ab^2

=b^2c-b^2a+bc^2-a^2b+ac^2-a^2c

=b^2(c-a)+b(c^2-a^2)+ac(c-a)

=(c-a)(b^2+ac)+b(c-a)(c+a)

=(c-a)(b^2+ac+bc+ba)

=(c-a)[b^2+bc+ac+ab]

=(c-a)[b(b+c)+a(b+c)]

=(c-a)(b+c)(b+a)

\(a,=\left(2x-5\right)\left(x+1\right)\\ b,=\left(x-10\right)\left(x+1\right)\\ c,=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)

`a, x^3 + y^3 + x + y`

`= (x+y)(x^2-xy+y^2)+x+y`

`= (x+y)(x^2-xy+y^2+1)`

`b, x^3 - y^3 + x -y`

`= (x-y)(x^2+xy+y^2)+x-y`

`= (x-y)(x^2+xy+y^2+1)`

`c, (x-y)^3 + (x+y)^3`

`= (x-y+x+y)(x^2-2xy+y^2 - x^2 + y^2 + x^2 + 2xy + y^2)`

`= (2x)(x^2 + 3y^2)`

`d, x^3 - 3x^2y + 3xy^2 - y^3 + y^2 - x^2`

`= (x-y)^3 + (y-x)(x+y)`

`=(x-y)(x^2+2xy+y^2-x-y)`

a: =(x+y)(x^2-xy+y^2)+(x+y)

=(x+y)(x^2-xy+y^2+1)

b: =(x-y)(x^2+xy+y^2)+(x-y)

=(x-y)(x^2+xy+y^2+1)

c: =x^3-3x^2y+3xy^2-y^3+x^3+3x^2y+3xy^2-y^3

=2x^3+6xy^2

d: =(x-y)^3+(y-x)(y+x)

=(x-y)[(x-y)^2-(x+y)]

a) = (x - 4y)(x + 1)

b) = (x - 3y)^2 - 2^2

= (x - 3y - 2)(x - 3y + 2)

c) = x^2(x + 3) - 7x(x + 3) + 9(x + 3)

= (x + 3)(x^2 - 7x + 9)

a: \(x^2-4xy+x-4y\)

\(=x\left(x-4y\right)+\left(x-4y\right)\)

\(=\left(x-4y\right)\left(x+1\right)\)

b: \(x^2-6xy+9y^2-4\)

\(=\left(x-3y\right)^2-4\)

\(=\left(x-3y-2\right)\left(x-3y+2\right)\)

( x + y + z)³ - x³ - y³ - z³=x³+y³+z³+3(a+b)(a+c)(b+c)- x³ - y³ - z³

                                              = 3(a+b)(b+c)(a+c)

\(\left(x+y-z\right)^3-x^3-y^3+z^3\)

\(=\left[\left(x+y\right)-z\right]^3-x^3-y^3+z^3\)

\(=\left(x+y\right)^3-z^3-3\left(x+y\right)z\left(x+y-z\right)-x^3-y^3+z^3\)

\(=x^3+y^3-z^3+3xy\left(x+y\right)-3\left(x+y\right)z\left(x+y-z\right)-x^3-y^3+z^3\)

\(=3xy\left(x+y\right)-3z\left(x+y\right)\left(x+y-z\right)\)

\(=3\left(x+y\right)\left[xy-z\left(x+y-z\right)\right]\)

\(=3\left(x+y\right)\left(xy-zx-yz+z^2\right)\)

\(=3\left(x+y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]\)

\(=3\left(x+y\right)\left(y-z\right)\left(x-z\right)\)

#\(Urushi\text{☕}\)

Áp dụng (a+b)3 = a3+b3+3ab(a+b), ta có:

(x+y+z)3-x3-y3-z3

=[(x+y)+z]3-x3-y3-z3

=(x+y)3+z3+3z(x+y)(x+y+z)-x3-y3-z3

=x3+y3+3xy(x+y)+z3+3z(x+y)(x+y+z)-x3-y3-z3

=3(x+y)(xy+xz+yz+z2)

=3(x+y)[x(y+z)+z(y+z)]

=3(x+y)(y+z)(x+z)

=(x+y-z-x)[(x+y-z)^2+x(x+y-z)+x^2]-(y-z)(y^2+yz+z^2)

=(y-z)(x^2+y^2+z^2+2xy-2xz-2yz+x^2+xy-xz+x^2-y^2-yz-z^2)

=(y-z)(3x^2+3xy-3xz-3yz)

=3(y-z)(x^2+xy-xz-yz)

=3(y-z)[x(x+y)-z(x+y)]

=3(y-z)(x+y)(x-z)

\(a,=y\left(y-2\right)\\ b,=3x\left(x^2-2x+1\right)=3x\left(x-1\right)^2\\ c,=\left(y-1\right)\left(27x^2+9x^3\right)=9x^2\left(x+3\right)\left(y-1\right)\\ d,=y\left(y^2-2y+1\right)=y\left(y-1\right)^2\\ e,=x\left(x^2+6x+9\right)=x\left(x+3\right)^2\\ f,=x\left(x^2-2xy+y^2\right)=x\left(x-y\right)^2\\ g,=\left(2-x\right)\left(x+1\right)\\ h,=\left(x-1\right)\left(3x-6\right)=3\left(x-1\right)\left(x-2\right)\)

a: =y(y-2)

b: \(=3x^2\left(x^2-2x+1\right)=3x^2\left(x-1\right)^2\)

d: \(=y\left(y^2-2y+1\right)=y\left(y-1\right)^2\)

x 3  +  y 3  +  z 3  – 3xyz = x + y 3  – 3xy(x + y) +  z 3  – 3xyz

      = [  x + y 3  +  z 3 ] - [ 3xy.(x+ y) + 3xyz]

      = [ x + y 3  +  z 3 ] – 3xy(x + y + z)

      = (x + y + z)[ x + y 2  – (x + y)z +  z 2 ] – 3xy(x + y + z)

      = (x + y + z)( x 2  + 2xy + y 2  – xz – yz + z 2  – 3xy)

      = (x + y + z)( x 2  +  y 2  +  z 2  – xy – xz - yz)

Ta có: ( x - y) z³ + ( y - z ) x³ + ( z - x ) y³  

= ( x - y ) z³ + ( y - z )x³ + ( z - y)y³ + ( y - x ) y³

= ( x - y ) ( z³ - y³ ) + ( y - z ) ( x³ - y³) 

= ( x - y ) ( z - y ) ( z² + zy + y² ) + ( y - z ) ( x - y) ( x² + xy + y² ) 

= ( x - y ) ( y - z ) ( x² + xy + y² - z² - zy - y²) 

= ( x - y ) ( y - z ) [ ( x² - z²) + ( xy - zy) ]

= ( x - y ) ( y - z ) [ ( x - z ) ( x + z ) + y ( x - z ) ]

= ( x - y ) ( y - z ) ( x - z ) ( x + y + z )